Description
Objective
This purpose of this exercise is to use numerical integration to solve Newton’s force law, thus
giving you the basic tools for modeling physical systems.
Prerequisites: Assignments 1, 2, 3, and 4. This activity uses Visual Python.
Introduction
Newton’s second law tells us that an object’s acceleration is directly proportional to the force we
apply to it. Measure the force and we can determine the subsequent motion. This means solving
the equation as a second order differential equation:
In principle, if we can write down a formula for the force we can solve the equation to get
acceleration as a function of time. In practice, however, very few physical systems allow for this
type of analytical solution.
Enter the computer. We can break this up into two first order differential equations like this
If we make dt sufficiently small, we can approximate the solution to these equations with the
following formula.
We can determine what will happen next provided we have some initial idea of what is
happening now. This is a task particularly well suited to a computer.
The Task
Given this mathematical framework, do the following:
1. Design an algorithm to calculate the 3-D projectile motion using this method. This is not
the same as using the analytical projectile motion formula you’ve seen in physics class,
but rather describing the force on an object and determining time step by time step what
is happening to the position and velocity. The user will provide initial position (x, y, z) and
initial velocities (vx, vy, vz), the time step, and how long the simulation will run. Your
dx
dt = v
dv
dt = F
m
F = ma
a = d2x
dt2 = F
m
v = v0 + adt
x = x0 + vdt
program will compute positions and velocities as a function of time and animate the
results, stopping the animation with the object hits the ground.
2. Write a code in Visual Python to perform this simulation, animating a ball being thrown.
In order to see the motion, you’ll need to provide some scenery (maybe a field or
backdrop of some kind, or some clouds, be as creative as you like!). Determine when to
start and stop the simulation so that you can visualize the parabolic motion of the object.
3. This should be identical to the projectile motion equation, but with this method we can
add in any forces we like. Air resistance is typically parameterized as
Where ρ is the air density, v is the velocity, CD is the drag coefficient (0.5 would be a
good estimate), and A is the cross sectional area of the object. Note that this vector
force points opposite of the direction of motion. Add this air resistance into your force
equation and rerun the simulation.
4. Finally, animate two objects, one with air resistance and one without, to show as a
comparison.
To hand in
Your final animation, showing the comparison between projectile motion with and without air
resistance.
F~D = 1
2
⇢vCDA~v
