Dice Problem, Part 1

I came across an interesting question on the “/r/learnmath” subreddit.  The poster describes two players, A and B, playing a game of dice. Each one has a die that has a specific side weighted. Each player rolls his die. If it lands on the weighted side, he gets the value of the side added to his score and rolls again. If it lands on any of the nonweighted sides, his turn is over. Whoever has the highest score wins. I found this question intriguing, and thought it would be a good opportunity for me to work on understanding infinite series, a weak area for me. Note that the original question states that if the first roll is a nonweighted side, the player gets that many points (instead of zero like on subsequent rolls). I’m going to ignore that rule and focus on the general pattern.

Problem: Each player has a die that is weighted on one side. He gets that many points and gets to roll again if it lands on the weighted side; if it lands on the unweighted side, his turn is over. More specifically, A’s weighted side is 3 and he has a 20% chance of rolling it; B’s weighted side is 2 and she has a 30% chance of rolling it. What is the chance that A will win (not tie)?

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Launch to Orbit, Part 3

In Part 2, I found the velocity required for circular orbit was

\displaystyle v = \sqrt{\frac{GM}{r}},

where G is the gravitational constant, M is the mass of the body being orbited, and r is the distance to the center of that mass. I tested this on the ISS’s orbit and found the velocity matched quite well. Now, I’m interested in looking at orbital periods, and calculating aspects of some common orbits.

Problem: Calculate the velocity and periods for some typical circular orbits.

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Launch to Orbit, Part 2

See Part 1, where I looked at how increasing horizontal launch velocities could lead to orbit. Now I want to explore this mathematically. In the Feynman Lectures on Physics, Vol. I, Ch. 7, Section 7–4 “Newton’s law of gravitation,” Feynman looks at the velocity required to achieve (circular) orbit. He looks at how far a projectile would fall in one second, and then looks at how fast it would have to be traveling horizontally to clear the surface and maintain the same altitude. I’d like to expand on his approach and see if I can find a general expression for the velocity.

Problem: Use the standard projectile motion equations to derive an expression for the velocity required to achieve circular orbit.

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Launch to Orbit, Part 1

As a way to understand orbital motion, I’ve seen the idea of “Newton’s cannonball,” illustrated below from Wikipedia:

Source: Wikipedia

In this thought experiment, a cannon at the top of a tall mountain fires a cannonball at increasing velocities, until eventually it moves so fast that it achieves orbit.

Problem: Model a projectile being launched into orbit in this manner, and visually explore the transition between falling to Earth and orbiting.

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Random Walk in One Dimension, Part 2

See Part 1. I’ve been exploring a random walk in one dimension, where on object moves either +1 or −1 at each step, at random. On average, it will end up back near where it started (distance = 0), but over time, the likely positions start to spread out. I previously graphed the outcome of 1000 trials for 2000 steps. Although I can see the trajectories start to spread out, I’d like to actually graph the distribution.

Problem: Graph the probability of an object ending up at different distances from the origin during a random walk.

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Kinetic Energy and Conservation of Energy, Part 2

See Part 1, where I used conservation of energy of a projectile to determine the formula for kinetic energy, Kmv². The aim was to see if I could derive a more intuitive understanding of this formula — especially why it depends on the square of the velocity. With a constant force and therefore constant acceleration, velocity increases linearly, by the same amount per second, so why does kinetic energy increase at increasingly large rates? (I should specify that I am using the nonrelativistic formula, so we assume speeds much lower than the speed of light.)

Problem: Can I understand the kinetic energy formula, Kmv², in a more intuitive manner?

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Kinetic Energy and Conservation of Energy, Part 1

I’ve been wanting to systematically go through a physics text, refreshing my knowledge and filling in gaps. I found The Feynman Lectures on Physics, freely available online, based on lectures from the great physicist and physics popularizer, Richard Feynman. As I read Chapter 4: “Conservation of Energy,” I wonder if I can derive the formula for kinetic energy, the energy an object has due to its motion. My primary motivation is that while it is relatively easy for me to grasp why the gravitational potential energy is U=mgh (that is, weight times height), I have a much harder time intuitively understanding why the kinetic energy should be proportional to the velocity squared: Kmv².

Problem: Given the formula for gravitational potential energy, can I use the principle of conservation of energy (of a projectile) to derive the equation for its kinetic energy?

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