A ball is an object

• Topic: objects, dynamics

• Task:

  1. Implement and test the simulation algorithm with just gravity.

  2. Also add air resistance and bouncing from the ground.

  3. Simulate the throw with different velocities or different physics.

• Template: ballistic.py

• Further reading:

  • https://en.wikipedia.org/wiki/Object-oriented_programming

  • https://en.wikipedia.org/wiki/Drag_(physics)

  • https://en.wikipedia.org/wiki/Coefficient_of_restitution

ballistic.py

class ballistic.Ball(x, y, vx, vy, mass=1.0, radius=0.1)

A center-of-mass model for a ball.

The model does not include rotation of the ball.

Parameters:

• x (float) – initial x-coordinate

• y (float) – initial y-coordinate

• vx (float) – initial velocity x-component

• vy (float) – initial velocity y-component

• mass (float) – ball mass

• radius (float) – ball radius, needed for bouncing

accelerate(dt)

Calculate the velocity of the ball after some time has passed.

The new velocity is calculated by

\[ \begin{align}\begin{aligned}v_x(t + \Delta t) = v_x(t) + \frac{F_x(t)}{m} \Delta t\\v_y(t + \Delta t) = v_y(t) + \frac{F_y(t)}{m} \Delta t.\end{aligned}\end{align} \]

The new velocity is saved in the properties of the ball.

Parameters:

dt (float) – timestep \(\Delta t\)

apply_air_resistance(drag)

Make air resistance act on the ball.

Adds the force \(\vec{F} = - \gamma m \vec{v}\).

Parameters:

drag (float) – air drag coefficient \(\gamma\)

apply_bounce(restitution)

Make the ball bounce from the ground (ground level is at y = 0).

Parameters:

restitution (float) – Coefficient for the elasticity of the bounce. 1 = completely elastic, 0 = completely inelastic.

apply_gravity(g)

Make gravity act on the ball.

Adds the force \(F_y = - m g\).

Parameters:

g (float) – gravitational acceleration \(g\)

move(dt)

Calculate the position of the ball after some has passed.

The new position is calculated by

\[ \begin{align}\begin{aligned}x(t + \Delta t) = x(t) + v_x(t) \Delta t + \frac{1}{2} \frac{F_x(t)}{m} (\Delta t)^2\\y(t + \Delta t) = x(t) + v_y(t) \Delta t + \frac{1}{2} \frac{F_y(t)}{m} (\Delta t)^2.\end{aligned}\end{align} \]

The new position is saved in the properties of the ball.

Parameters:

dt (float) – timestep \(\Delta t\)

record_position()

Save the position of the ball in memory.

reset_forces()

Forget the forces from a previous time.

ballistic.animate(ball)

Animate the motion of the ball.

Parameters:

ball (Ball) – the ball whose trajectory will be drawn

ballistic.draw_animation_frame(frame, xs, ys)

Draws the system at a certain time. Used for animation.

Parameters:

• frame (int) – index of the time step to draw

• xs (list) – array of ball x-coordinates

• ys (list) – array of ball y-coordinates

ballistic.draw_motion_blur_image(ball)

Draws the trajectory of the ball as a series or points.

Parameters:

ball (Ball) – the ball whose trajectory will be drawn

ballistic.draw_trajectory(ball)

Draws the trajectory of the ball as a line.

Parameters:

ball (Ball) – the ball whose trajectory will be drawn

ballistic.main(v_start, angle)

Main program.

Creates a ball, throws it at an angle and simulates the throw.

Parameters:

• v_start (float) – initial speed

• angle (float) – throwing angle in degrees, measured from the horizontal

ballistic.run_simulation(ball, g, drag, restitution, dt, simulation_time, recording_dt)

Run a dynamic simulation.

The simulation runs repeating the following steps:

• Forces are applied on the ball using e.g. Ball.apply_gravity().

• The ball is moved with Ball.move().

• The velocity is updated with Ball.accelerate().

The trajectory is saved in the Ball object.

Note

This function is incomplete!

Parameters:

• ball (Ball) – the moving ball

• g (float) – acceleration due to gravity

• drag (float) – air resistance factor (between 0 and 1)

• restitution (float) – impact elasticity factor (between 0 and 1)

• dt (float) – time step \(\Delta t\)

• simulation_time (float) – total simulation time

• recording_dt (float) – time between trajectory recording