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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Random Walk in One Dimension, Part 1

As I continue reading the Feynman Lectures on Physics, I’m intrigued by the discussion of the random walk in Chapter 6: “Probability,” Section 6–3: “The random walk.” Feynman discusses a random walk in one dimension, where at each step an object moves one unit either forward or backward at random. In Figure 6–5, he draws a graph tracing the distance moved over 30 steps, for 3 trials. I remember seeing graphs like this in my textbooks when I was in school, studying them and trying to appreciate them. And I realize that now technology has advanced so much that I can explore these further on my home computer.

Problem: Graph a random walk in one dimension, following more trials over longer periods.

Continue reading “Random Walk in One Dimension, Part 1”

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?

Continue reading “Kinetic Energy and Conservation of Energy, Part 2”