Ballistics

Ballistic Motion

Ballistic motion is motion that is affected by gravity. Imagine a ball thrown up into the air that rises to its maximum height and then comes back down. For simplicity, we'll ignore winds and friction with the air.

To write a program that simulates ballistic motion, we'll start with the simple, vector motion of a ball model.

Notes	Code and Results
Motion without gravity. We have a ball moving upward (in the y-direction) at 5 m/s, so its velocity vector is: $\vec{v} = <0, 5, 0>$ Notice that: Line 1: `scene.range = 10` Sets a constant field of view so we don't automatically zoom in or out as the ball moves. The core of the model are lines 24 and 25: `ball.pos.x = ball.pos.x + ball.v.x * dt` These equations for the position comes from the fact that, as we all know, velocity is the change in distance over time. Considering only motion in the x direction (v_x): $v_x = \dfrac{\Delta x}{\Delta t}$ We can write the change in distance (Δx) as the difference in the old and new positions: $v_x = \dfrac{x_{new} - x_{old}}{\Delta t}$ Therefore, if we know the velocity (v), the time step (Δt), and the starting position (x_old), then we can solve this equation to find the new position (x_new). $x_{new} = x_{old} + v_x \cdot \Delta t \tag{1}$ This could be read as: the new position is equal to the old position plus the change in position. There is no motion in the x direction, but the same general equation applies in y, so: `ball.pos.y = ball.pos.y + ball.v.y * dt`	scene.range = 10 ball = sphere(radius=0.5) ball.v = vector(0, 5, 0) ball.arrow = arrow(pos = ball.pos, axis = ball.v, shaftwidth=0.2) ball.vx_arrow = arrow(pos = ball.pos, axis = vec(ball.v.x, 0, 0), color=color.red, shaftwidth=0.2) ball.vy_arrow = arrow(pos = ball.pos + ball.vx_arrow.axis, axis = vec(0, ball.v.y, 0), color=color.yellow, shaftwidth=0.2) dt = 0.05 while True: sleep(dt) sleep(1) # update ball position based on velocity vector ball.pos.x = ball.pos.x + ball.v.x * dt ball.pos.y = ball.pos.y + ball.v.y * dt #update positions of the arrows ball.arrow.pos = ball.pos ball.vx_arrow.pos = ball.pos ball.vy_arrow.pos = ball.pos + ball.vx_arrow.axis # update directions of arrows ball.arrow.axis = ball.v ball.vx_arrow.axis = vec(ball.v.x, 0, 0) ball.vy_arrow.axis = vec(0, ball.v.y, 0) The result should look like:
Adding gravity (acceleration) Acceleration is the change in velocity (Δv) with time (Δt) so: $a = \dfrac{v_{new} - v_{old}}{\Delta t}$ So for each timestep, we'll solve this equation for the new velocity (v_new ): $v_{new} = v_{old} + a \cdot \Delta t \tag{1}$ The acceleration due to gravity (g) is: $a = g = -9.8 \, \text{m/s}^2$ So, for vertical, ballistic motion we can write the equation as: $v_{new} = v_{old} + g \cdot \Delta t \tag{2}$ We can add this as a single line to the previous program, before we calculate the new position. Line 23: `ball.v.y = ball.v.y - 9.8 * dt`	scene.range = 10 ball = sphere(radius=0.5) ball.v = vector(0, 10, 0) ball.arrow = arrow(pos = ball.pos, axis = ball.v, shaftwidth=0.2) ball.vx_arrow = arrow(pos = ball.pos, axis = vec(ball.v.x, 0, 0), color=color.red, shaftwidth=0.2) ball.vy_arrow = arrow(pos = ball.pos + ball.vx_arrow.axis, axis = vec(0, ball.v.y, 0), color=color.yellow, shaftwidth=0.2) dt = 0.05 while True: sleep(dt) #sleep(1) # update velocity based on gravitational acceleration ball.v.y = ball.v.y - 9.8 * dt # update ball position based on velocity vector ball.pos.x = ball.pos.x + ball.v.x * dt ball.pos.y = ball.pos.y + ball.v.y * dt #update positions of the arrows ball.arrow.pos = ball.pos ball.vx_arrow.pos = ball.pos ball.vy_arrow.pos = ball.pos + ball.vx_arrow.axis # update directions of arrows ball.arrow.axis = ball.v ball.vx_arrow.axis = vec(ball.v.x, 0, 0) ball.vy_arrow.axis = vec(0, ball.v.y, 0)
Add a floor Add a floor to your model so the ball bounces.

Notes

Code and Results

Motion without gravity.

We have a ball moving upward (in the y-direction) at 5 m/s, so its velocity vector is:

$\vec{v} = <0, 5, 0>$

Notice that:

Line 1:
```
scene.range = 10
```
Sets a constant field of view so we don't automatically zoom in or out as the ball moves.

The core of the model are lines 24 and 25:

ball.pos.x = ball.pos.x + ball.v.x * dt

These equations for the position comes from the fact that, as we all know, velocity is the change in distance over time. Considering only motion in the x direction (v_x):

$v_x = \dfrac{\Delta x}{\Delta t}$ We can write the change in distance (Δx) as the difference in the old and new positions:

$v_x = \dfrac{x_{new} - x_{old}}{\Delta t}$ Therefore, if we know the velocity (v), the time step (Δt), and the starting position (x_old), then we can solve this equation to find the new position (x_new).

$x_{new} = x_{old} + v_x \cdot \Delta t \tag{1}$ This could be read as: the new position is equal to the old position plus the change in position.

There is no motion in the x direction, but the same general equation applies in y, so:

ball.pos.y = ball.pos.y + ball.v.y * dt

scene.range = 10

ball = sphere(radius=0.5)

ball.v = vector(0, 5, 0)

ball.arrow = arrow(pos = ball.pos, axis = ball.v, shaftwidth=0.2)
ball.vx_arrow = arrow(pos = ball.pos,
                      axis = vec(ball.v.x, 0, 0),
                      color=color.red,
                      shaftwidth=0.2)
ball.vy_arrow = arrow(pos = ball.pos + ball.vx_arrow.axis,
                      axis = vec(0, ball.v.y, 0),
                      color=color.yellow,
                      shaftwidth=0.2)

dt = 0.05

while True:
    sleep(dt)
    sleep(1)

    # update ball position based on velocity vector
    ball.pos.x = ball.pos.x + ball.v.x * dt
    ball.pos.y = ball.pos.y + ball.v.y * dt

    #update positions of the arrows
    ball.arrow.pos = ball.pos
    ball.vx_arrow.pos = ball.pos
    ball.vy_arrow.pos = ball.pos + ball.vx_arrow.axis
    # update directions of arrows
    ball.arrow.axis = ball.v
    ball.vx_arrow.axis = vec(ball.v.x, 0, 0)
    ball.vy_arrow.axis = vec(0, ball.v.y, 0)

The result should look like:

Adding gravity (acceleration)

Acceleration is the change in velocity (Δv) with time (Δt) so:

$a = \dfrac{v_{new} - v_{old}}{\Delta t}$ So for each timestep, we'll solve this equation for the new velocity (v_new ):

$v_{new} = v_{old} + a \cdot \Delta t \tag{1}$ The acceleration due to gravity (g) is:

$a = g = -9.8 \, \text{m/s}^2$ So, for vertical, ballistic motion we can write the equation as:

$v_{new} = v_{old} + g \cdot \Delta t \tag{2}$ We can add this as a single line to the previous program, before we calculate the new position.

Line 23:
```
ball.v.y = ball.v.y - 9.8 * dt
```

scene.range = 10

ball = sphere(radius=0.5)

ball.v = vector(0, 10, 0)

ball.arrow = arrow(pos = ball.pos, axis = ball.v, shaftwidth=0.2)
ball.vx_arrow = arrow(pos = ball.pos,
                      axis = vec(ball.v.x, 0, 0),
                      color=color.red,
                      shaftwidth=0.2)
ball.vy_arrow = arrow(pos = ball.pos + ball.vx_arrow.axis,
                      axis = vec(0, ball.v.y, 0),
                      color=color.yellow,
                      shaftwidth=0.2)

dt = 0.05

while True:
    sleep(dt)
    #sleep(1)

    # update velocity based on gravitational acceleration
    ball.v.y = ball.v.y - 9.8 * dt

    # update ball position based on velocity vector
    ball.pos.x = ball.pos.x + ball.v.x * dt
    ball.pos.y = ball.pos.y + ball.v.y * dt

    #update positions of the arrows
    ball.arrow.pos = ball.pos
    ball.vx_arrow.pos = ball.pos
    ball.vy_arrow.pos = ball.pos + ball.vx_arrow.axis
    # update directions of arrows
    ball.arrow.axis = ball.v
    ball.vx_arrow.axis = vec(ball.v.x, 0, 0)
    ball.vy_arrow.axis = vec(0, ball.v.y, 0)

Add a floor

Add a floor to your model so the ball bounces.