# Motion Along a Curve, Part 3

See Part 1 and Part 2, where I discuss trying to find equations of motion for a ball rolling along a track defined by an arbitrary function y(x). So far, I’ve worked out my general approach, and tested it on the very simple case of an inclined line. I do want to tackle some complex functions, but first I want to summarize the method, and to incorporate some changes I learned as I tried my first solution.

I had initially stated I wanted to start with x(0) = 0, to keep it simple. But I didn’t end up needing this restriction. My equation for velocity used the initial yi, which we got from plugging in the initial value of x. Also, we used the x(0) = 0 to find the constant after integrating. But these happened later in the process.

I also had started with an initial velocity of zero. That did make a difference. But looking back on part 1, I don’t think it would complicate the equation too much, and if it’s zero it will be an extra term to just drop out. Let’s go back to the conservation of energy equation, and keep vi in this time.

$\displaystyle U_i + K_i = U + K$

$\displaystyle mgy_i + \frac{1}{2}m{v_i}^2 = mgy + \frac{1}{2}mv^2$

$\displaystyle gy_i - gy = \frac{1}{2}(v^2 - {v_i}^2)$

$\displaystyle 2g(y_i - y) = v^2 - {v_i}^2$

$\displaystyle 2g(y_i - y) + {v_i}^2 = v^2$

$\displaystyle v = \sqrt{2g(y_i-y)+{v_i}^2} \blacktriangleleft$

I’m leaving the ± in this time. Strictly speaking, I’m not treating this as the magnitude of the vector, since magnitudes must be positive. Rather, I want to consider a velocity vector that can point ether forwards or backwards along the direction of the curve. I’m going to allow motion in both directions, not just forward.

Recall the graph showing components of the velocity vector:

Now I want to find the x-component. As I discovered last time, I don’t need to bother with the y-component — once I find an equation for x(t), I can use that directly to obtain y(t). The x-component will be

$\displaystyle v_x = v \frac{1}{\sqrt{(y')^2+1}}$

$\displaystyle v_x = \pm \sqrt{\frac{2g(y_i - y) + {v_i}^2}{(y')^2+1}}$

$\displaystyle \frac{dx}{dt} = \pm \sqrt{\frac{2g(y_i - y) + {v_i}^2}{(y')^2+1}} \blacktriangleleft$

Since I will have y and y′ in terms of x, I would need to rearrange to solve the differential equation. Let’s see how far I can take the general case:

$\displaystyle \pm \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = dt$

$\displaystyle \pm \int \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = \int dt$

$\displaystyle \pm \int \sqrt{\frac{(y')^2+1}{2g(y_i-y)+{v_i}^2}} \, dx = t \blacktriangleleft$

where I did not include a constant of integration on the right side, since it can be absorbed into the constant that the left integral will produce.

So, the general approach should be as follows: Given our equation y(x), find y′(x). Plug in those expressions, plug in the initial velocity, and plug in the initial height y[x(0)]. Integrate, and solve for x in terms of t to get x(t), then plug that into y(x) to get y(t). I’ll test if this approach can actually work in the next post.

# Motion Along a Curve, Part 2

I would like to find equations for the position of a small ball rolling along a curve given by an equation, rather than a flat incline. In Part 1, I introduced the problem and set up a test case. Now let’s try to actually make some headway.

Last time, we showed that

$\displaystyle v = \sqrt{2g(y_i - y)}$

where v is the magnitude of the velocity vector v, g is the acceleration due to gravity, and y is the height. I note that if y>yi, then the quantity inside the radical is negative and v will not be real — which makes sense, since the ball should never be able to roll higher than its starting position. (Recall that I am starting with an initial velocity of zero). Since we’re assuming that the ball rolls along the function, its direction will be given by the derivative of y with respect to x. I do see a potential problem with notation arising. The value of y depends on x (by definition, since that’s how I’m defining the curve). But x and y both depend on t if we’re thinking about the motion of the ball. I want to define our variables carefully. We should think of x as a function x(t), and y as a function of x, so y(x) = y[x(t)]. I am going to reserve the prime notation y′ specifically to refer to the derivative with respect to x.

$\displaystyle y' = \frac{dy}{dx}$

We know that

$\displaystyle \mathbf{v} = \langle r, \angle\theta \rangle$

The magnitude will be v. If we take the tangent line to the curve and sketch a triangle, we can see that for a change in x of 1, y changes by y′, so tan θ = y′/1. I illustrated this below with an arbitrary curve.

$\displaystyle \mathbf{v} = \langle v, \angle \arctan y' \rangle$

Let’s now find the component vectors vx and vy, where of course vx = r cos θ and vy = r sin θ.

$\displaystyle \mathbf{v} = \langle v \cos \arctan y', v \sin \arctan y' \rangle$

There’s actually no need for these arctangents. If I go back to the earlier sketched triangle, I can see that the hypotenuse is $\sqrt{(y')^2 +1}$ and so I can read the cosine and sine directly off that triangle (1 over the hypotenuse and y’ over the hypotenuse, respectively):

$\displaystyle \mathbf{v} = \left\langle v \frac{1}{\sqrt{(y')^2 + 1}}, v \frac{y'}{\sqrt{(y')^2 +1}} \right\rangle$

$\displaystyle \mathbf{v} = \frac{v}{\sqrt{(y')^2+1}}\langle 1, y' \rangle$

Let’s substitute in our expression for the magnitude, v:

$\displaystyle \mathbf{v} = \sqrt{\frac{2g(y_i-y)}{(y')^2+1}} \langle 1, y' \rangle \blacktriangleleft$

I believe that this should be the general solution, given the constraints (ball must roll along the curve, uniform gravity, no friction, starting x and v of 0). Let’s pause to make some observations. The expression in the numerator is the magnitude of the velocity which we found in the last post. The potential/kinetic energy relationship is obvious, with the product of g and the change in height from potential energy, and the times two and the square root being inverses from kinetic energy. The <1, y′> ensures the proper direction, and the denominator adjusts for the length of that direction vector (it basically makes it a unit vector).

This equation is so far pretty straightforward. However, in order to find equations for x and y in terms of time t, we’d have to plug in the expression for y and y′, then integrate the velocity. Let’s try this with what should be a very simple example, the incline (line) from the prior entry:

$\displaystyle y = -\frac{x\sqrt{3}}{3} + 1$

$\displaystyle y' = -\frac{\sqrt{3}}{3}$

$\displaystyle y_i = y(0) = 1$

Plugging these into our equation for v gives:

$\displaystyle \mathbf{v} = \sqrt{\frac{2g(y_i-y)}{(y')^2+1}} \langle 1, y' \rangle$

$\displaystyle \mathbf{v} = \sqrt{\frac{2g[1-(-\frac{x\sqrt{3}}{3}+1)]}{(-\frac{\sqrt{3}}{3})^2+1}} \left\langle 1, -\frac{\sqrt{3}}{3} \right\rangle$

I am starting to regret setting my initial angle to π/6. There was no benefit to choosing a “nice” value for the angle; it would have been better to choose a slope with an integer value. However, it’s too late to go back and change it now. Let me see if I can simplify this.

$\displaystyle \mathbf{v} = \sqrt{\frac{2g\frac{x\sqrt{3}}{3}}{\frac{1}{3}+1}} \left\langle 1, -\frac{\sqrt{3}}{3} \right\rangle$

$\displaystyle \mathbf{v} = \sqrt{\frac{2gx\sqrt{3}}{\frac{4}{3}\cdot 3}} \left\langle 1, -\frac{\sqrt{3}}{3} \right\rangle$

$\displaystyle \mathbf{v} = \sqrt{\frac{gx\sqrt{3}}{2}} \left\langle 1, -\frac{\sqrt{3}}{3} \right\rangle$

Let’s split this into its component vectors, and switch to magnitudes:

$\displaystyle \mathbf{v} = \mathbf{v_x} + \mathbf{v_y}$

$\displaystyle \mathbf{v} = v_x \boldsymbol{\hat{\i}} + v_y \boldsymbol{\hat{\j}}$

$\displaystyle v_x = \sqrt{\frac{gx\sqrt{3}}{2}}$

and

$\displaystyle v_y = -\sqrt{\frac{gx\sqrt{3}}{2}} \cdot\frac{\sqrt{3}}{3} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

Let’s now try integrating the first one:

$\displaystyle \frac{dx}{dt} = \sqrt{\frac{gx\sqrt{3}}{2}}$

$\displaystyle \frac{1}{\sqrt{x}} \, dx = \sqrt{\frac{g\sqrt{3}}{2}} \, dt$

$\displaystyle \int x^{-\frac{1}{2}} dx = \int \sqrt{\frac{g\sqrt{3}}{2}} \, dt$

$\displaystyle \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C_1 = t \sqrt{\frac{g\sqrt{3}}{2}} + C_2$

$\displaystyle 2\sqrt{x} = t \sqrt{\frac{g\sqrt{3}}{2}} + C_3$

$\displaystyle \sqrt{x} = t \sqrt{\frac{g\sqrt{3}}{8}} + C_3$

This is looking promising. Before I square both sides, let’s deal with the constant. Since we’re starting at a position x=0 at t=0,

$\displaystyle 0 = 0 + C_3$

so clearly the constant is zero. (This won’t be so simple for the y-component.)

$\displaystyle \sqrt{x} = t \sqrt{\frac{g\sqrt{3}}{8}}$

I will square both sides, realizing that I might be introducing extraneous solutions. In fact, t cannot be negative, since the square root of x is nonnegative.

$\displaystyle x(t) = \frac{g\sqrt{3}}{8} t^2, \, t\geq 0 \blacktriangleleft$

Before tackling the y-direction, let’s test this answer. Let’s plug in our value from the last post for t, for the time when it reaches the x-axis.

$\displaystyle x\left(\sqrt{\frac{8}{g}}\right) = \frac{g\sqrt{3}}{8} \left(\sqrt{\frac{8}{g}}\right)^2 = \frac{g\sqrt{3}}{8} \cdot \frac{8}{g} = \sqrt{3}$

This is indeed the value of x at the base of the triangle (recall height 1, width √3, hypotenuse 2 for a 30-60-90 triangle)! Now let’s find y. Let’s start with our equation for the y-component of the velocity:

$\displaystyle v_y = -\sqrt{\frac{gx\sqrt{3}}{6}}$

This one is going to be a little more complicated.

$\displaystyle \frac{dy}{dt} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

I already see a complication. My equation has x, not y. We of course have our starting equation for y in terms of x, and it would be easy to solve that for x and substitute it in. However, if we want to try this on other functions, especially those that aren’t injective (that is, where multiple values of x might result in the same value of y — in a parabola, or sine curve for instance), then solving will be difficult. I think it would be better to try to use the chain rule.

$\displaystyle \frac{dy}{dx} \frac{dx}{dt} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

We already found y′ above:

$\displaystyle -\frac{\sqrt{3}}{3} \frac{dx}{dt} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

I just realized something. We already have $\frac{dx}{dt}$. It was the vx we started with to find x(t) above!

Wait. That expression is in terms of x, and we have no y’s left. Substituting that in will just result in an identity.

$\displaystyle -\frac{\sqrt{3}}{3} \sqrt{\frac{gx\sqrt{3}}{2}} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

$\displaystyle -\sqrt{\frac{gx\sqrt{3}}{6}} = -\sqrt{\frac{gx\sqrt{3}}{6}}$

This is bad news, because I really don’t think that solving the starter equation for y will be a practical solution for most functions. But if we don’t have a y or a dy, then we’re not going to get an equation for y.

Scratch that. There’s no need to do any of this! We just derived an equation x(t). We already know y(x). So we can find y[x(t)]!

$\displaystyle y(x) = -\frac{x\sqrt{3}}{3} + 1$

$\displaystyle x(t) = \frac{g\sqrt{3}}{8}t^2$

So, plugging in the equation for x in,

$\displaystyle y(t) = -\frac{\sqrt{3}}{3} \left(\frac{g\sqrt{3}}{8}\right) t^2 + 1$

$\displaystyle y(t) = -\frac{g}{8}t^2 + 1, \, t\geq 0 \blacktriangleleft$

This looks right. It’s of a similar form to our equation for x(t), but with a factor of -1 instead √3 (which follows from our 30-60-90 triangle, and y is decreasing. And it has a +1 to raise the initial height. Just to be sure, let’s check it with our test value for t:

$\displaystyle y\left(\sqrt{\frac{8}{g}}\right) = -\frac{g}{8}\left(\sqrt{\frac{8}{g}}\right)^2 + 1 = -1 + 1 = 0$

And that’s correct, because we specifically picked the test case for when the ball reaches the x-intercept. So our final solution is

$\displaystyle x(t) = \frac{g\sqrt{3}}{8} t^2, \,t\geq0 \blacktriangleleft$

$\displaystyle y(t) = -\frac{g}{8}t^2 + 1, \, t\geq 0 \blacktriangleleft$

The method works! In future explorations, it would be interesting to try this for a more complicated starter equation.

The restriction t≥0 is interesting. If it weren’t there, the equations for the motion of the ball would still be valid, since it would represent the ball being launched backwards up the slope, coming to rest for an instant at (0,1) at t=0, then rolling back down the slope. I’m not surprised that it is there, though, because my method explicitly assumed that the ball’s velocity vector points in the direction <1, y′>, which points in the forward x direction. Let’s trace back the restriction on t. It comes from where it times a constant equals √x, which must be positive. Why did we have √x, as opposed to ±√x? That came from the equation for the magnitude of the velocity v. That in turn came from the fact that K = ½mv2. Should we have kept the negative solution? No, because v represents the magnitude, and the magnitude is nonnegative. Still, a negative v would correspond to a vector pointing in the opposite direction, when put into our component equation. I believe that those solutions should be allowed. I could justify it by saying that v was not the magnitude, but rather ±magnitude, to allow the vector to point in either direction. Now that I think about it, this will occur for every problem we attempt with this method. Since the laws of physics and therefore the equations of motion should be symmetric over time, this means that if the ball is starting at rest at time = 0, we could “run the film in reverse” to see what path it took to get to the starting point. This is an interesting development, and I wonder if expanding the method in this manner would allow the ball to roll backwards at other times, too — say if it were to roll down a valley, then up the other side. If we allow these “negative magnitudes,” might we be able to get an equation which expresses the motion as it rolls back down into the valley and back up the original side?

# Motion Along a Curve, Part 1

Earlier this week, I saw an intriguing problem suggested, one involving motion along a curve. The idea is this: in elementary mechanics, we learn about how an object will move when placed on an inclined surface. If we restrict motion to two dimensions, this surface can be represented by a line. With gravity as the propelling force, can we find a general approach to deriving its motion along an arbitrary curve? In other words, given a function f(x), can we find functions describing the position (and therefore velocity and acceleration) over time?

Here are my initial thoughts. One, I am sure that this type of problem has been analyzed before. That doesn’t matter; I am going to try figuring it out myself. Two, I think we can use conservation of energy to determine the object’s velocity at any point. However, that will depend on knowing its height, and I don’t know how this approach would handle the object flying up off a short hill, or dropping away from a cliff. So three, I am going to restrict the object’s motion to the curve — consider it a track, rather than a 1D surface. That means that four, I can use the derivative to find the direction, but that means the object can only move forward. Five, I have to ignore friction. My conservation of energy approach will require that all energy be potential or kinetic; I won’t know hot to deal with losses to friction. Six, I am using constant downward gravitation. Seven, I am starting with initial position x=0 and initial velocity of 0. I don’t believe this method requires it, but I am already concerned about the complexity of the math. And seven, I am planning to obtain a velocity equation and integrate to find position, but I don’t know if the equation will have an analytic solution (or if I will know how to integrate it).

These are some significant limitations, of course. If this approach works, I suspect that it could be expanded to deal with several of these. For instance, we could keep in terms for initial position and velocity. Gravity or whichever force could be represented by a vector field rather than a constant force; this would make the potential energy term much more complex. I also wonder if I could incorporate friction by adding in an extra term, but I think this would turn the equation into a more complex differential equation.

Let’s start with a simple case that we can solve using conventional means, so we’ll have an answer to check later. I can also use this to check my conservation of energy approach. Let’s have the object rolling down an incline of 30° (π/6), starting at a height of 1. The ball will roll down to the right (I find it more intuitive to imagine a rolling ball rather than a frictionless object sliding, especially if the surface is curved). The equation for this surface will be

$\displaystyle y(x) = -\frac{x\sqrt{3}}{3}+1 \, \bullet$

where the slope is -tan(π/6).

You can see that it would form a right triangle with height 1, length √3, and hypotenuse 2. The slope is therefore 1/√3, or √3/3. Perhaps it would have been better to use an incline of 60° (π/3) so that the slope would be -√3, but I like this one better. Let’s try the conservation of energy approach to see what the velocity would be when the ball reaches the bottom (that is, y = 0). The total energy E is the sum of the potential energy U and the kinetic energy K. This should remain constant, so

$\displaystyle E_i = E_f$

$\displaystyle U_i + K_i = U_f + K_f$

We know that potential energy is given by

$\displaystyle U = mgh$

assuming constant gravitation g (the acceleration due to gravity), with mass m and height h; and kinetic energy is given by

$\displaystyle K = \frac{1}{2}mv^2$

where v is the magnitude of the velocity vector v. I confess I am not very proficient with vectors, but clearly the ball is moving in two dimensions and we will need both components in the x and y directions. I am going to try to not be sloppy but to carefully think about what we mean by v or v every time I write it.

We assume that the initial velocity, and therefore kinetic energy, is zero.

$\displaystyle U_i + K_i = U_f + K_f$

$\displaystyle mgh_i + \frac{1}{2}m\cdot0^2 = mgh_f + \frac{1}{2}mv_f^2$

The height is y, and we can divide both sides by the nonzero mass m:

$\displaystyle gy_i = gy_f + \frac{1}{2}v_f^2$

Let’s solve for v:

$\displaystyle 2g(y_i-y_f)=v_f^2$

$\displaystyle v_f = \sqrt{2g(y_i-y_f)} \blacktriangleleft$

This should be the general case, so if this checks out, we can use this equation to develop the method. As a check, if the units of g are m/s² and y is in m, the units will be m/s, appropriate for velocity. Also note that this is simply the magnitude of the velocity vector (the speed); the actual vector could point in any direction if all we know is conservation of energy. That’s why I earlier constrained the ball to move along the curve — that will give us our direction. Plugging in our initial and final values of y, we get

$\displaystyle v_f = \sqrt{2g(1-0)} = \sqrt{2g}$

Next, to verify this solution, let’s solve using the traditional free-body approach. Gravity applies a force downwards of W=mg. This can be split into two components, one perpendicular to the surface mg cos(θ), and one along the surface mg sin(θ). The perpendicular force will be balanced out by the equal and opposite normal force, leaving a force of mg sin(θ). This is simple motion in one dimension, and let me introduce a variable s to represent its position along the surface (so I don’t generate confusion with our x’s and y’s). Think of it as placing a tape measure from the bottom of the incline up to the starting point. We know that for a given constant acceleration a,

$\displaystyle v = \int \! a \, dt = at + v_0$

$\displaystyle s = \int \! v \, dt = \int \! (at+v_0) \, dt = \frac{1}{2}at^2 + v_0 t + s_0$

By Newton’s second law, F = ma, so a = g sin(θ) and will be negative, since it’s pointing towards decreasing s (since this is motion in one dimension, we can use the sign to expression direction rather than need vectors). Initial velocity is zero, and initial position is 2 (the distance along the incline, if the bottom is at s=0).

$\displaystyle 0 = -\frac{1}{2}g \sin(\frac{\pi}{6}) \cdot t^2 + 0t + 2$

$\displaystyle \frac{1}{2} g \left( \frac{1}{2}\right)t^2 = 2$

$\displaystyle \frac{gt^2}{4} = 2$

$\displaystyle t^2 = \frac{8}{g}$

$\displaystyle t = \pm \sqrt{\frac{8}{g}}$

I can discard the negative solution, since I’m interested in the behavior starting from time = 0, not how it might have been launched before to come to a temporary stop at time = 0. Let’s plug this time to get our velocity from v = at (since the initial velocity is zero):

$\displaystyle v = at = -g\left(\frac{1}{2}\right)\sqrt{\frac{8}{g}} = - \sqrt{\frac{g^2}{4}}\sqrt{\frac{8}{g}} = -\sqrt{2g}$

This velocity is negative since it points downwards along the s direction, along the incline. Its magnitude will be the absolute value which is what we obtained earlier using the conservation of energy method. Armed with this success, and with some test values, we’ll be ready to work on the actual problem in the next post.

Edit: I added a graph.

# Simple Harmonic Motion

I came across an interesting approach to a problem involving a spring the other day, and it made me wonder if I could derive the equations for simple harmonic motion. The basic principle of a spring is Hooke’s law, that the spring exerts a restoring force proportional to its displacement from equilibrium position: F = –kx, where k is the spring constant representing the stiffness of the spring. I wondered if I could derive the formulas for its motion, and where the sinusoidal components would appear from, since clearly there are no trigonometric functions in this basic equation. To begin with, from Newton’s second law, F = ma:

$\displaystyle ma = -kx$

$\displaystyle a = -\frac{kx}{m}$

We know that acceleration is the derivative of velocity and velocity is the derivative of position:

$\displaystyle a = \frac{dv}{dt}$

and

$\displaystyle v = \frac{dx}{dt} .$

We can rewrite these as a dt = dv and v dt = dx, and integrate. However, our acceleration equation has a in terms of position x, not time t, so I would prefer to integrate with respect to x (as at this point I don’t know how x depends on t — that’s what we’re trying to derive!).

$\displaystyle a \, dt = dv$

$\displaystyle a \left( \frac{dx}{v} \right) = dv$

$\displaystyle a \, dx = v \, dv$

This equation is well-known and could have been used as a starting point, but it’s nice to derive it from the most basic principles. We can now substitute our expression in for a and integrate both sides.

$\displaystyle -\frac{kx}{m} \, dx = v \, dv$

$\displaystyle \int \! \left( -\frac{kx}{m} \right) \, dx = \int \! v \, dv$

$\displaystyle -\frac{k}{m} \int \! x \, dx = \int \! v \, dv$

$\displaystyle -\frac{k}{m} \left( \frac{x^2}{2} + C_1 \right) = \frac{v^2}{2} + C_2$

Let’s solve this for v:

$\displaystyle \frac{v^2}{2} + C_2 = -\frac{k}{m} \left( \frac{x^2}{2} + C_1 \right)$

$\displaystyle \frac{v^2}{2} + C_2 = -\frac{kx^2}{2m} + C_3$

$\displaystyle \frac{v^2}{2} = -\frac{kx^2}{2m} + C_4$

$\displaystyle v^2 = -\frac{kx^2}{m} + C_5$

$\displaystyle v = \pm \sqrt { -\frac{kx^2}{m} + C_5 }$

There’s still no sign of sine appearing anywhere (I know; I couldn’t resist). I’m also a bit uncomfortable with the plus/minus sign, but velocity can be both positive and negative, so I don’t think that discarding the negative solution is justified. I want to integrate again to remove the velocity, but we face a similar problem as before: we have velocity in terms of position x, not time t. We can rearrange the differential again:

$\displaystyle v = \frac{dx}{dt}$

to get v on the dx side:

$\displaystyle dt = \frac{dx}{v} .$

Let’s substitute in our expression for v, and integrate:

$\displaystyle dt = \frac{dx}{\pm \sqrt { -\frac{kx^2}{m} + C_5 } }$

$\displaystyle \int dt = \int \! \frac{dx}{\pm \sqrt { -\frac{kx^2}{m} + C_5 } }$

This is not pretty, and I’m not sure how to integrate the right side. I believe I can pull out the plus/minus as a constant.

$\displaystyle t + C_6 = \pm \int \! \frac{dx}{\sqrt { -\frac{k}{m} x^2 + C_5 } }$

I had to consult a table of integrals for assistance with this integration. Of course, it can be solved with trignometric substitution! All of a sudden, we have a hint of where a sine function will appear. We need to get the radical in the form $\sqrt{a^2 - b^2 x^2 }$:

$\displaystyle \sqrt{C_5 - \frac{k}{m} x^2 }$

$\displaystyle a = \sqrt{C_5} = C_7$

$\displaystyle b= \sqrt{\frac{k}{m}}$

Now, we set

$\displaystyle x = \frac{a}{b} \sin \theta :$

$\displaystyle x = C_7\sqrt{\frac{m}{k}}\sin \theta$

Let’s now take our original integral back, and start substituting.

$\displaystyle t + C_6 = \pm \int \! \frac{dx}{\sqrt { C_5 -\frac{k}{m} x^2 } }$

$\displaystyle t + C_6 = \pm \int \! \frac{dx}{\sqrt { {C_7}^2 -\frac{k}{m} x^2 } }$

$\displaystyle t + C_6 = \pm \int \! \frac{dx}{\sqrt { {C_7}^2 -\frac{k}{m} \left( C_7\sqrt{\frac{m}{k}}\sin \theta \right) ^2 } }$

This expression is complicated, but I can already see how it will simplify out.

$\displaystyle t + C_5 = \pm \int \! \frac{dx}{\sqrt{ {C_7}^2 - \frac{k}{m} \cdot {C_7}^2 \cdot \frac{m}{k} \cdot \sin^2 \theta} }$

$\displaystyle t + C_5 = \pm \int \! \frac{dx}{\sqrt{ {C_7}^2 - {C_7}^2 \sin^2 \theta } }$

$\displaystyle t + C_5 = \pm \int \! \frac{dx}{\sqrt{{C_7}^2 (1 - \sin^2 \theta)} }$

$\displaystyle t + C_5 = \pm \frac{1}{C_7} \int \! \frac{dx}{\sqrt{1 - \sin^2 \theta} }$

Let’s convert the dx to a dθ:

$\displaystyle x = C_7\sqrt{\frac{m}{k}}\sin \theta$

$\displaystyle \frac{dx}{d\theta} = C_7\sqrt{\frac{m}{k}}\cos \theta$

$\displaystyle dx = C_7\sqrt{\frac{m}{k}}\cos \theta \, d\theta$

Substituting this back in yields:

$\displaystyle t + C_5 = \pm \frac{1}{C_7} \int \! \frac{C_7\sqrt{\frac{m}{k}}\cos \theta \, d\theta }{\sqrt{1 - \sin^2 \theta} }$

$\displaystyle t + C_5 = \pm \frac{C_7}{C_7}\sqrt{\frac{m}{k}}\int \! \frac {\cos \theta}{\sqrt{1 - \sin^2 \theta}} d\theta$

$\displaystyle t + C_5 = \pm \sqrt{\frac{m}{k}}\int \! \frac{\cos \theta}{\sqrt{\cos^2 \theta}} d\theta$

$\displaystyle t + C_5 = \pm \sqrt{\frac{m}{k}}\int \! \frac{\cos \theta}{\cos \theta} d\theta$

$\displaystyle t + C_5 = \pm \sqrt{\frac{m}{k}}\int d\theta$

$\displaystyle t + C_5 = \pm \theta \sqrt{\frac{m}{k}} + C_8$

$\displaystyle t + C_9 = \pm \theta \sqrt{\frac{m}{k}}$

Remembering our equation for x in terms of θ:

$\displaystyle C_7\sqrt{\frac{m}{k}}\sin \theta = x$

$\displaystyle \sqrt{\frac{m}{k}}\sin \theta = C_{10}x$

$\displaystyle \sin \theta = C_{10}x\sqrt{\frac{k}{m}}$

$\displaystyle \theta = \arcsin \left(C_{10}x\sqrt{\frac{k}{m}} \right)$

Finally, we can substitute this back in for θ and solve for x:

$\displaystyle t + C_9 = \pm \left[ \arcsin \left(C_{10}x\sqrt{\frac{k}{m}} \right) \right] \sqrt{\frac{m}{k}}$

$\displaystyle \pm t\sqrt{\frac{k}{m}} + C_{11} = \arcsin \left(C_{10}x\sqrt{\frac{k}{m}} \right)$

The plus-minus is driving me crazy, but I can’t justify dropping it — even though I want to.

$\displaystyle \sin \left( \pm t\sqrt{\frac{k}{m}}+ C_{11} \right) = C_{10}x\sqrt{\frac{k}{m}}$

Actually, I think I can get rid of it now! Since sin(-x) = -sin(x) = sin(x+π), I can incorporate the +π into the constant!

$\displaystyle C_{10}x\sqrt{\frac{k}{m}} = \sin \left( t\sqrt{\frac{k}{m}} + C_{12} \right)$

$\displaystyle x = C_{13} \sin \left( t\sqrt{\frac{k}{m}} + C_{12} \right)$

Let’s rewrite this equation using the conventional symbols for the constants.

$\displaystyle x = A \sin \left( t\sqrt{\frac{k}{m}} - \varphi \right)$

There it is. The basic sine wave equation is readily apparent, with the constants representing the amplitude and the phase shift, and the radical representing the angular frequency. The fourth possible constant (a “+ D”) is absent, and I wondered if I had accidentally dropped a constant somewhere — since that entire family of curves should be solutions. That is, x should be able to oscillate around 1, -5, whatever. But the reason for its absence is our starting point of Hooke’s law, F=-kx, which assumes that the equilibrium point is x=0. We would have had to start with F=-k(xxeq) to have had that last constant.

It’s pretty neat to see the equation emerge from just four basic equations: Hooke’s law F = –kx, Newton’s second law F=ma, and the definitions of velocity and acceleration v=x′ and a=v′!

# Snail Problem, Part 5

I hope no one is tired of snails yet. In the last several posts, I tackled the following problem: a snail crawls along a twig of length L0 at 1 cm/d, and the twig grows (along its entire length) at 2 cm/d. Will the snail reach the end, and when? In parts 1, 2, and 3, I solved this problem, and then in part 4 I turned to the general case, leaving the snail’s velocity and twig growth velocity unspecified.

We found that the snail’s position, x, can be given by the following equation, where g is the growth rate of the twig, s is the (inherent) velocity of the snail, and t is the time:

$\displaystyle x = \frac{s}{g}(gt+L_0)\ln\frac{gt+L_0}{L_0}.$

We want to find the time t where the snail reaches the end of the twig. Since the length of the twig is gt+L0, we set x equal to this.

$\displaystyle x = gt+L_0$

Substituting our equation for x gives

$\displaystyle \frac{s}{g}(gt+L_0)\ln\frac{gt+L_0}{L_0}=gt+L_0$.

To solve this for t, we start by dividing both sides by gt+L0 (note that this is the twig length, and is only applicable for nonzero twig lengths).

$\displaystyle \frac{s}{g}\ln\frac{gt+L_0}{L_0}=1$

We can multiply both sides by g/s (note that the snail velocity cannot be zero).

$\displaystyle \ln\frac{gt+L_0}{L_0}=\frac{g}{s}$

Exponentiating both sides gives

$\displaystyle e^{\ln\frac{gt+L_0}{L_0}}=e^{\frac{g}{s}}$

$\displaystyle \frac{gt+L_0}{L_0}=e^{\frac{g}{s}}$

$\displaystyle gt+L_0 = L_0 e^{\frac{g}{s}}$

$\displaystyle gt = L_0 e^{\frac{g}{s}}-L_0$

$\displaystyle gt = L_0(e^{\frac{g}{s}}-1)$

And finally, the time is

$\displaystyle t_f = \frac{L_0}{g}\left(e^{g/s}-1\right) \blacksquare$

So the time it takes is the initial length of the twig divided by its growth rate (which incidentally is the amount of time it would have taken to grow to its start position) times the quantity e to the power of the ratio of the twig and snail rates minus 1.

Interestingly, this will be finite no matter how small s is. If the snail is extremely slow, then the exponent g/s will be very large, but as long as s is positive, the snail will eventually reach the end of the twig — no matter how fast the twig grows!

Let’s also see how long the twig will be when the snail reaches the end. Recall that the length of the twig is

$\displaystyle L(t) = gt+L_0$

We can substitute our time above in for t:

$\displaystyle L(t_f) = g\left[\frac{L_0}{g}\left(e^{g/s}-1\right)\right] +L_0$

$\displaystyle L(t_f) = L_0\left(e^{g/s}-1\right)+L_0$

$\displaystyle L(t_f) = L_0\left[(e^{g/s}-1)+1\right]$

And finally, the length is

$\displaystyle L(t_f) = L_0 e^{g/s}\, \blacksquare$

# Snail Problem, Part 4

In the last three posts (see Snail Problem parts 1, 2, and 3), I tackled the following problem: a snail crawls along a twig of length L0 at 1 cm/d, and the twig grows 2 cm/d. Will the snail reach the end, and when?

The answer is yes, and it will reach the end in

$\displaystyle t = (e^2-1)\cdot\frac{L_0}{2}.$

I’m interested in solving this now for the general case, leaving the twig growth and snail velocities unspecified. Fellow blogger Ken Roberts has already solved this, but I still want to try my hand at it. I’ll follow his example and use g for the growth rate of the twig (I’m already using t for time), s for the inherent velocity of the snail, and I’ll continue using L0 for the initial length of the twig (and L for its length at any time).

The length of the twig L is

$\displaystyle L(t)= gt+L_0$,

since it grows at g cm/d and its grown for t days. The velocity at any point x is length is the fraction of the length of the twig,

$\displaystyle \frac{x}{L(t)} = \frac{x}{gt + L_0}$

times the twig’s overall growth rate, g:

$\displaystyle \frac{gx}{gt + L_0}$.

The net velocity v of the snail is the sum of the growth of the twig at that point (which we just calculated) and its own inherent velocity, s:

$\displaystyle v = \frac{gx}{gt + L_0} + s$

Of course, v is the derivative of the snail’s position x with respect to time. Remember that L0, g, and s are constants.

$\displaystyle \frac{dx}{dt} = \frac{gx}{gt + L_0} + s$

This is a linear first-order differential equation.

$\displaystyle \frac{dx}{dt} - \frac{g}{gt + L_0}x = s$

The way to solve this is to multiply by the following integrating factor (see part 2 for an explanation):

$\displaystyle \mu(t) = e^{\int \left(-\frac{g}{gt + L_0}\right) \,dt}$

$\displaystyle \mu(t) = e^{-\int g\cdot\frac{1}{gt + L_0}\,dt}$

$\displaystyle \mu(t) = e^{-\ln|gt+L_0| + c_1}$

I actually might prefer doing this the other way (the longer method from the previous post), since integrating inside an exponent is inconvenient. Using sum and product rules for exponents:

$\displaystyle \mu(t) = e^{c_1}{\left(e^{ln |gt+L_0|}\right)}^{-1}$

$\displaystyle \mu(t) = c_2(gt+L_0)^{-1}$

I got rid of the absolute value because the quantity gt + L0 will always be positive, since it’s the length of the twig. And more generally, we can select a quantity for the constant that would have the appropriate sign. We just need one possible solution for this integrating factor, so I’ll select the one where the constant is equal to one.

$\displaystyle \mu(t) = (gt+L_0)^{-1}$

Now we take this and multiply it by our prior differential equation.

$\displaystyle (gt+L_0)^{-1}\left(\frac{dx}{dt} - \frac{g}{gt + L_0}x\right) = (gt+L_0)^{-1}s$

$\displaystyle (gt+L_0)^{-1}\frac{dx}{dt}-g(gt+L_0)^{-2}x = \frac{s}{gt+L_0}$

The whole point of this diversion was to get the left side of the equation into a form that is the inverse of the product rule.

$\displaystyle \frac{d}{dt}\left[(gt+L_0)^{-1}x\right] = \frac{s}{gt+L_0}$

Integrating both sides with respect to t gives

$\displaystyle (gt+L_0)^{-1}x = \int\frac{s}{gt+L_0}\,dt$

$\displaystyle (gt+L_0)^{-1}x = s\int\frac{1}{gt+L_0}\,dt$

We need to set up the inverse of the chain rule.

$\displaystyle (gt+L_0)^{-1}x = \frac{s}{g}\int\frac{g}{gt+L_0}\,dt$

$\displaystyle (gt+L_0)^{-1}x = \frac{s}{g}\left(\ln|gt+L_0| + c_3\right)$

We can multiply through by s/g, and define a new constant. Also, I’ll remove the absolute value, since I’m only interested in positive t.

$\displaystyle (gt+L_0)^{-1}x=\frac{s}{g}\ln(gt+L_0) + c_4$

$\displaystyle x = \left[\frac{s}{g}\ln(gt+L_0) + c_4\right](gt+L_0)$

We can find the constant by noting that when t=0, x=0:

$\displaystyle 0 = \left(\frac{s}{g}\ln L_0 + c_4\right)(L_0)$

Dividing both sides by L0 gives

$\displaystyle 0 = \frac{s}{g}\ln L_0 + c_4$

$\displaystyle c_4 = -\frac{s\ln L_0}{g}$

Substituting this back into our equation for x:

$\displaystyle x = \left[\frac{s}{g}\ln(gt+L_0) -\frac{s\ln L_0}{g}\right](gt+L_0)$

$\displaystyle x = \frac{s}{g}\left[\ln(gt+L_0)-\ln L_0\right](gt+L_0)$

Let’s combine the natural logarithms and rearrange the terms for clarity:

$\displaystyle x= \frac {s}{g}(gt+L_0)\ln\frac{gt+L_0}{L_0}$

That’s it! Noting that the quantity gt+L0 represents the current twig length, L, we can say that

$\displaystyle x = \frac{s}{g}L\ln\frac{L}{L_0}$.

In other words, at any given time, the position of the snail is equal to the ratio of the snail’s inherent velocity to the twig growth velocity times the current twig length times the natural logarithm of the ratio of the twig’s current length to its original length. Keeping it in this form, using the general case rather than s=1 and g=2, certainly makes the formula more intuitive. I’ll solve for the time it takes to reach the end of the twig in the next post.

# Snail Problem, Part 3

This is a continuation of my attempt to solve the following problem: A snail crawls along a twig of length L0 at 1 cm/d, and the twig grows (along its entire length) at 2 cm/d. Will the snail reach the end, and when? You can see part 1 where I successfully set up the differential equation and unsuccessfully tried to solve it, and part 2 where I did manage to solve it.

At the end of the last post, I had solved for the equation that describes the position of the snail over time,

$\displaystyle x = \left(t+\frac{L_0}{2}\right)\ln\left(\frac{2t}{L_0}+1\right)$.

Also, recall that the length of the twig is

$\displaystyle L = 2t + L_0$

since it grows at 2 cm/d. We want to find the time t when these two are equal.

$\displaystyle 2t + L_0 = \left(t+\frac{L_0}{2}\right)\ln\left(\frac{2t}{L_0}+1\right)$

[Edit: I missed a very obvious step at this point, so the next few steps will be unhelpful. Free free to skip ahead to the next edit.] We need to isolate the natural logarithm.

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = \frac{2t+L_0}{t+\frac{L_0}{2}}$

Exponentiating both sides gives

$\displaystyle e^{\ln\left(\frac{2t}{L_0}+1\right)} = e^{\frac{2t+L_0}{t+\frac{L_0}{2}}}$

$\displaystyle \frac{2t}{L_0}+1 = e^{(2t+L_0)(t+\frac{L_0}{2})^{-1}}$

Well, the left side is great, but the right side is a mess.

$\displaystyle \frac{2t}{L_0}+1 = \left[e^{(t+\frac{L_0}{2})^{-1}}\right]^{(2t+L_0)}$

If I bring the 2tL0 term out, I could split it up.

$\displaystyle \frac{2t}{L_0}+1 = \left[e^{(t+\frac{L_0}{2})^{-1}}\right]^{(2t+L_0)}$

$\displaystyle \frac{2t}{L_0}+1 = \left[e^{(t+\frac{L_0}{2})^{-1}}\right]^{2t}\left[e^{(t+\frac{L_0}{2})^{-1}}\right]^{L_0}$

This was unhelpful and I have no idea how to simplify this.

[Edit: At this point I get back on track.] Let’s go back to the last simple equation.

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = \frac{2t+L_0}{t+\frac{L_0}{2}}$

I should get rid of this complex fraction, even if it means making the numerator more complex.

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = \frac{2t+L_0}{t+\frac{L_0}{2}}\cdot\frac{2}{2}$

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = \frac{4t+2L_0}{2t+L_0}$

Oh, it’s so obvious now. I don’t know how I missed realizing that this fraction would simplify.

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = \frac{2(2t+L_0)}{2t+L_0}$

$\displaystyle \ln\left(\frac{2t}{L_0}+1\right) = 2$

I feel a bit foolish. Now, let’s exponentiate both sides.

$\displaystyle e^{\ln\left(\frac{2t}{L_0}+1\right)} = e^2$

That’s much simpler.

$\displaystyle \frac{2t}{L_0}+1 = e^2$

$\displaystyle \frac{2t}{L_0} = e^2 - 1$

And finally, the long-awaited answer,

$\displaystyle t = (e^2-1)\cdot\frac{L_0}{2}$.

Here are my initial observations: It seems “elegant” enough to be a solution to this problem. I like that it depends on e. I also notice that the time it takes is directly proportional to the initial length of the twig. I’m a bit surprised, as the snail’s net speed is not constant but rather increases as it progresses. Also, it’s apparent in retrospect that the first factor, $\left(t+\frac{L_0}{2}\right)$ in the equation for the snail’s position over time,

$\displaystyle x = \left(t+\frac{L_0}{2}\right)\ln\left(\frac{2t}{L_0}+1\right)$,

is half the length of the twig at that time, L(t):

$\displaystyle x(t) = \frac{1}{2}L(t)\ln\left(\frac{2t}{L_0}+1\right)$.

I don’t have an intuitive understanding of what the other quantity could represent, nor do I know if the 2 and 1 have any connection to the twig growth and snail inherent velocity. It would be interesting to repeat this but leave all the quantities unspecified.

# Snail Problem, Part 2

I read through some more of my college differential equations textbook and I think I am ready to further tackle the snail problem (see yesterday’s post). In short, a snail crawls along a twig of length L0 at 1 cm/d, and the twig grows 2 cm/d. Will the snail reach the end, and when?

I got as far as this differential equation, which I believe is correct:

$\displaystyle \frac{dx}{dt} = \frac{2x}{2t + L_0} + 1$

I read about a technique to solve linear first-order differential equations last night, so I’m going to try it here. First we put this into standard form:

$\displaystyle \frac{dx}{dt} -\frac{2}{2t + L_0}\cdot x = 1.$

Since we already have an x’ and an x, if only they were multiplied by another function μ and μ’ respectively, we would have an inverse of the product rule (μx)’ = (μx’ + μ’x). Of course we can’t multiply different parts of the equation by different functions; we’ll have to multiply the entire equation by the same function μ(t). This will put a μ term in front of the x’, so we have to select a μ that, when multiplied by the function in the second term, will result in μ’. Let’s take a detour to find this function.

$\displaystyle \mu' = \mu\left( -\frac{2}{2t + L_0} \right)$

This is a separable differential equation; we can get the μ terms on one side and the t terms on the other. (Incidentally, I don’t think we could have done that with the original equation, or I would have.)

$\displaystyle \frac{d\mu}{dt} = \mu\left( -\frac{2}{2t + L_0} \right)$

$\displaystyle \frac{1}{\mu}\,d\mu = -\frac{2}{2t+L_0}\,dt$

I just realized we divided both sides by μ. I hope that was “allowed.” Of course we would have lost any solutions where μ = 0, though I’m not sure of the significance of that. In any case, I can’t stop now. Let’s integrate:

$\displaystyle \int \frac{1}{\mu}\,d\mu = \int \left( -\frac{2}{2t+L_0} \right) \,dt$

The left side is easy; it’s a standard rule.

$\displaystyle \ln |\mu|= \int \left( -\frac{2}{2t+L_0} \right) \,dt$

The right side is essentially the same; we need to use the inverse of the chain rule.

$\displaystyle \ln |\mu| = - \int \left( \frac{1}{2t+L_0} \right) (2) \,dt$

$\displaystyle \ln |\mu| = - \ln | 2t+L_0 | + c_1$

Exponentiating both sides yields

$\displaystyle e^{ \ln |\mu| } = e^{- \ln | 2t+L_0 | + c_1}$

$\displaystyle |\mu| = {e^{c_1}} {\left( e^{ln |2t+L_0|} \right)}^{-1}$

Setting $c_2 := e^{c_1}$

$\displaystyle |\mu| = c_2 {|2t+L_0|}^{-1}$

$\displaystyle \mu = \pm \frac{c_2}{2t+L_0}$

I think we can absorb the ± into the constant (though I’m not positive).

$\displaystyle \mu = \frac{c_3}{2t+L_0}$

Actually, we need a solution, not all solutions, so I’ll set the constant equal to 1. So dropping the ± was fine.

$\displaystyle \mu = {(2t+L_0)}^{-1}$

Armed with μ, we should now be ready to return to our earlier work. We’ll take our previous equation, in standard form,

$\displaystyle \frac{dx}{dt} -\frac{2}{2t + L_0}\cdot x = 1$

and multiply both sides by the function μ:

$\displaystyle {(2t+L_0)}^{-1} \cdot \frac{dx}{dt} - 2{(2t + L_0)}^{-2} \cdot x = {(2t+L_0)}^{-1}$

We should now be able to a use the inverse of the product rule.

$\displaystyle \frac{d}{dt} \left[{(2t+L_0)}^{-1} x \right] = {(2t+L_0)}^{-1}$

Integrating both sides gives

$\displaystyle {(2t+L_0)}^{-1} x = \int {(2t+L_0)}^{-1}\,dt$

We’ll need to use the inverse of the chain rule again, so let’s set it up.

$\displaystyle \frac{x}{2t+L_0} = \frac{1}{2}\int 2{(2t+L_0)}^{-1}\,dt$

$\displaystyle \frac{x}{2t+L_0} = \frac{1}{2} \left(\ln |2t+L_0| + c_4\right)$

$\displaystyle \frac{x}{2t+L_0} = \frac{ \ln |2t+L_0|}{2} + c_5$

Multiplying both sides by 2tL0 gives

$\displaystyle x = \left( \frac{\ln |2t+L_0|}{2} + c_5 \right) (2t+L_0)$

Wow, we already solved for x. I wasn’t expecting it so soon. The solution is a little more complex than I’d expected, and I suspect I made a mistake somewhere. However, we should keep going. Let’s find the constant. The snail starts at position x = 0 at time t = 0, so

$\displaystyle 0 = \left( \frac{\ln |0+L_0|}{2} + c_5 \right) (0 + L_0)$

L0 is positive, so we don’t need the absolute value.

$\displaystyle 0 = \left( \frac{\ln L_0}{2} + c_5 \right)L_0$

$\displaystyle 0 = \frac{L_0 \ln L_0}{2} + c_5 L_0$

$\displaystyle c_5 L_0 = - \frac{L_0 \ln L_0}{2}$

$\displaystyle c_5 = - \frac{\ln L_0}{2}$

This is actually promising. Substituting c5 back into the equation we solved for x,

$\displaystyle x = \left( \frac{\ln |2t+L_0|}{2} + c_5 \right) (2t+L_0)$

$\displaystyle x = \left( \frac{\ln |2t+L_0|}{2} - \frac{\ln L_0}{2} \right) (2t+L_0)$

$\displaystyle x = \left( \frac{\ln |2t+L_0| - \ln L_0}{2} \right) (2t+L_0)$

I want to get rid of the absolute value. I think we can. The natural logarithm is only defined for positive numbers (at least without using complex numbers), so the absolute value allows our function to be defined for all nonzero arguments (just like $1/x$). If I remove the absolute value function, then the logarithm and therefore the equation will not be defined for 2t + L0 < 0, or when t < –L0/2. Since negative time is meaningless in this example (the snail started at time t = 0), I don’t mind discarding those times. (As an aside, I haven’t kept the units, but the 2 in the denominator is actually 2 cm/d — the speed at which the twig grows. The time –L0/2 corresponds to the time at which the twig had length zero. Clearly it would be meaningless in this problem to consider times before this.)

Back to the problem:

$\displaystyle x = \left[ \frac{\ln (2t+L_0) - \ln L_0}{2} \right] (2t+L_0)$

$\displaystyle x = [\ln (2t+L_0) - \ln L_0] \cdot \frac{2t+L_0}{2}$

$\displaystyle x = \ln \frac{2t+L_0}{L_0} \cdot \left(t + \frac{L_0}{2}\right)$

$\displaystyle x = \left(t+\frac{L_0}{2}\right) \ln \left(\frac{2t}{L_0} + 1\right)$

And that’s it! That is the equation for the distance the snail has crawled, x, at any given time t. We haven’t actually answered the question of if the snail will reach the end and when that would be, but we can save that for the next post.

# Snail Problem

A friend mentioned this problem a while back, and while I was able to use a computer to get an accurate numerical solution, I really want to find an analytical solution using math. The problem is that it will require differential equations, which I no longer remember. Edit: I haven’t yet been able to solve this problem, but I’m preserving my initial attempts below. See Part 2 where I made more progress.

The problem is this: a snail starts crawling along a twig, from one end towards the other. The snail can crawl at 1 cm/d. But the twig is growing at 2 cm/d. The twist is that the twig is adding material along its entire length, so the snail will be carried forward a bit, too, depending on where it is. Will the snail ever reach the end? Let’s let L0 be the initial length of the twig.

Let’s say the snail crawls at 1 cm/d and the twig grows at 2 cm/d. The length of the twig at time t would be

$\displaystyle L(t) = L_0 + 2t$.

For any point along the twig, the velocity will be proportional to the distance along the length of the twig, so at a distance x

$\displaystyle v_t(x, t) = \frac{x}{L_0 +2t} \cdot 2 = \frac{2x}{L_0 + 2t}$

Let’s double-check this: the beginning of the twig is $x = 0$:

$\displaystyle v_t(0,t) = \frac{2 \cdot 0}{L_0 + 2t} = 0$.

And at the end of the twig, $x = L_0 + 2t$:

$\displaystyle v_t(L_0+2t,t) = \frac{2(L_0+2t)}{L_0+2t} = 2$.

The velocity of the snail will include both its movement along the twig (at 1 cm/d) and the growth of the twig. It will be therefore

$\displaystyle v_s(x,t) = \frac{2x}{L_0+2t} + 1$

Since the velocity is the derivative of its position,

$\displaystyle \frac{dx}{dt} = \frac{2x}{L_0 + 2t} + 1$

and we want to know if and when $x(t) = L(t)$. In other words, solve for t:

$\displaystyle x(t) = L_0 + 2t$

Taking the derivative with respect to t,

$\displaystyle \frac{d}{dt} x(t) = \frac{d}{dt} (L_0 + 2t)$

$\displaystyle \frac{dx}{dt} = 2$

Note that this makes sense, at the time t when the snail reaches the end, … wait, no it doesn’t, that’s not correct. Since the functions are only equal at a specific time t, I can’t differentiate them and expect them to be equal.

Going back:

$\displaystyle \frac{dx}{dt} = \frac{2x}{L_0 + 2t} + 1$

Is this an linear differential equation? I hope so, since I’m hoping that will make it easier to solve (though at this point, I don’t remember how to solve linear or nonlinear differential equations.

$\displaystyle \frac{dx}{dt} - \frac{2}{L_0 + 2t} \cdot x = 1$.

I think I got it into the correct form. I don’t yet know what to do from here, so I’ll read on and also refresh my memory of differentiation rules.

I actually don’t think we’ll need complex differential equation techniques. We should be able to just integrate this entire equation with respect to t. I think I see why linear differential equations are easier to solve.

$\displaystyle \int ( \frac{dx}{dt} - \frac{2}{L_0 + 2t} \cdot x )\,dt = \int 1\,dt$

By the sum rule,

$\displaystyle \int \frac{dx}{dt} \,dt - \int \frac{2}{L_0+2t} \cdot x\,dt = \int 1\,dt$.

I spoke too soon. The first term is easy to integrate and the last is trivial, but I don’t know how to integrate the second.

$\displaystyle x + c_1 - \int \frac{2}{L_0+2t} \cdot x\,dt = t + c_2$

I can rearrange the terms and combine the constants

$\displaystyle \int \frac{2}{L_0+2t} \cdot x\,dt = x - t + c_3$

and I’m now stuck.

OK, I think we might be able to use integration by parts, using the formula

$\displaystyle \int u\,dv = uv - \int v\,du$.

If I set

$\displaystyle u =\frac{2}{L_0+2t}$

and set

$\displaystyle dv = x\,dt$

then to find v, we need to integrate:

$\displaystyle v = \int x\,dt$

$\displaystyle v = \frac{x^2}{2} + c_4$.

And to find du, we need to take the derivative of u. I’m just going to use the product rule, since I’m rusty at this and I don’t remember the quotient rule.

$\displaystyle du = \frac{d}{dt}\frac{2}{2t+L_0} = \frac{d}{dt}[2\cdot (2t+L_0)^{-1}]$

$\displaystyle du = \left(\frac{d}{dt}2\right) \cdot \frac{1}{2t+L_0} + 2 \cdot \frac{d}{dt}(2t+L_0)^{-1}$

$\displaystyle du = 0 + 2 \cdot (-1)\cdot(2t+L_0)^{-2}\cdot 2$

I’m going to keep going, but I just realized that I am not going to be able to integrate $\int v\,du$.

$\displaystyle du = -\frac{4}{(2t+L_0)^2}$

Shoot.

$\displaystyle uv - \int v\,du = \frac{2}{2t+L_0}\cdot \left(\frac{x^2}{4}+c_4 \right) + \int \left(\frac{x^2}{4}+c_4 \right) \left[\frac{4}{(2t+L_0)^2}\right]\,dt$.

This is a disaster.

I will have to study this more and return to this problem later.

# Individual Action is Not Enough

The Canadian chapter of the World Wildlife Federation produced this cool video/commercial:

Most people try, to at least some degree, to take steps to help the environment. And these small changes, when summed across the whole population, are significant. But still, the collective action of individuals can only do so much — government and industry need to be on board, too. Unfortunately, in the United States, leadership from the federal government has been lacking (and at times, actively impedes) so state and local governments and industry have had to take their own steps. There is some hope, though, that this will change when the Obama administration takes office (I hope to write more on this in a later post).