Projectile Motion Simulation

About this experiment

Launch a ball at any angle and speed and watch the parabola it traces. The simulation splits the motion into a constant horizontal velocity and a vertical motion with constant downward acceleration, so students can see why the path is a parabola and how each variable changes the range and height.

Learning goals

Decompose projectile motion into independent horizontal and vertical motions
Predict the maximum height, range and time of flight from the launch conditions
Explain why 45° gives the longest range on level ground and why it changes with launch height
Compare the same launch under the gravity of different planets

How to use it

1Set the launch angle and initial speed with the sliders, or drag from the launcher on the stage to aim.
2Press Launch. The blue and green arrows show the horizontal and vertical velocity components during flight.
3Change one variable at a time and launch again. Up to five earlier paths stay on screen for comparison.
4Switch the gravity preset to the Moon or Jupiter and compare the new trajectory with the faded Earth path.

The model

x = v_{0} cos θ t

Horizontal position: constant velocity

y = h_{0} + v_{0} sin θ t - \frac{1}{2} g t^{2}

Vertical position: constant acceleration g downward

H = h_{0} + \frac{v _{0}^{2} sin ^{2} θ}{2 g}

Maximum height

R = \frac{v _{0}^{2} sin 2 θ}{g} (h_{0} = 0)

Range on level ground

Assumptions

Air resistance is ignored, so the horizontal velocity never changes.
Gravity is uniform and points straight down.
The ball is treated as a point mass and the flight ends when it reaches the ground (y = 0).

Try this in class

Q1Find two different angles that give the same range. What do they add up to?
Q2Raise the launch height to 20 m. Is 45° still the angle with the longest range?
Q3Double the initial speed. By what factor does the range change, and why?
Q4At the highest point, which velocity component is zero and which is not?

Frequently asked questions

Why does 45° give the maximum range?

On level ground the range is proportional to sin 2θ, which reaches its largest value of 1 when 2θ = 90°, that is θ = 45°.

Does a heavier ball land farther?

Not in this model. Without air resistance every object falls with the same acceleration g, so mass does not appear in any of the equations.

Why do real balls fall short of the prediction?

Air drag slows the ball in both directions. The effect grows with speed, so fast launches fall shorter than the ideal parabola predicts.