Projectile motion

Launch a ball with a chosen speed, angle and height, with or without air resistance.

Kinematics — Describing motion: position, velocity and acceleration, without asking what caused it.

Open it in The Lab The equations behind it

Experiments already set up on it

Textbook launch (no drag)
Ideal projectile on level ground. Range depends on v₀² sin 2θ / g.
- Independence of horizontal and vertical motion
- Range vs launch angle
- Energy conservation
Tennis ball with air resistance
A 58 g ball with quadratic drag. Compare with the vacuum prediction.
- Effect of drag on range and optimal angle
- Energy dissipated by non-conservative forces
Launch from a cliff
Horizontal launch from 30 m. Flight time depends only on height.
- Flight time independent of horizontal speed
- Quadratic relation between height and time
On the Moon
Same launch, g = 1.62 m/s². How much farther does it go?
- Dependence of range on g
Bouncy ball
A ball with e = 0.7 dropped from 10 m. Each bounce returns to e² of the previous height.
- Coefficient of restitution from bounce heights
- Geometric decay of energy: Eₙ = e^2n E₀
Trampoline course
Two bounce pads: a tilted pad launches the ball forward onto a powered one.
- Reflection of velocity about a surface normal
- Energy added by a powered pad

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)
Horizontal positionx (m)
Heighty (m)
Horizontal velocityvₓ (m/s)
Vertical velocityvᵧ (m/s)
Speed|v| (m/s)
Horizontal accelerationaₓ (m/s^2)
Vertical accelerationaᵧ (m/s^2)
Kinetic energyEₖ (J)
Potential energyEₚ (J)
Mechanical energyE (J)
Drag force|F_d| (N)
Bounces so farn_b

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Launch speedv₀ (m/s)
Launch angleθ (°)
Launch heighth₀ (m)
Massm (kg)
Gravitational field strengthg (m/s^2)
Linear drag coefficientb (kg/s)
Drag coefficientC_d
Radiusr (m)
Air densityρ (kg/m^3)
Ball bouncinesse
Stop after bouncesmaxBounces

The equations behind it

What projectile motion is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Kinematics

$v = \frac{Δ s}{Δ t}$
Average velocity
$a = \frac{Δ v}{Δ t}$
Average acceleration
$v = u + a t$
Velocity after a time — Constant acceleration only.
$s = u t + \frac{1}{2} a t^{2}$
Displacement from a time
$v^{2} = u^{2} + 2 a s$
Velocity from a displacement
$s = \frac{( u + v )}{2} t$
Displacement from the average velocity

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency

Experiments already set up on it

Textbook launch (no drag)

Ideal projectile on level ground. Range depends on v₀² sin 2θ / g.

Independence of horizontal and vertical motion
Range vs launch angle
Energy conservation

Tennis ball with air resistance

A 58 g ball with quadratic drag. Compare with the vacuum prediction.

Effect of drag on range and optimal angle
Energy dissipated by non-conservative forces

Launch from a cliff

Horizontal launch from 30 m. Flight time depends only on height.

Flight time independent of horizontal speed
Quadratic relation between height and time

On the Moon

Same launch, g = 1.62 m/s². How much farther does it go?

Dependence of range on g

Bouncy ball

A ball with e = 0.7 dropped from 10 m. Each bounce returns to e² of the previous height.

Coefficient of restitution from bounce heights
Geometric decay of energy: Eₙ = e^2n E₀

Trampoline course

Two bounce pads: a tilted pad launches the ball forward onto a powered one.

Reflection of velocity about a surface normal
Energy added by a powered pad

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)

Horizontal positionx (m)

Heighty (m)

Horizontal velocityvₓ (m/s)

Vertical velocityvᵧ (m/s)

Speed|v| (m/s)

Horizontal accelerationaₓ (m/s^2)

Vertical accelerationaᵧ (m/s^2)

Kinetic energyEₖ (J)

Potential energyEₚ (J)

Mechanical energyE (J)

Drag force|F_d| (N)

Bounces so farn_b

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Launch speedv₀ (m/s)

Launch angleθ (°)

Launch heighth₀ (m)

Massm (kg)

Gravitational field strengthg (m/s^2)

Linear drag coefficientb (kg/s)

Drag coefficientC_d

Radiusr (m)

Air densityρ (kg/m^3)

Ball bouncinesse

Stop after bouncesmaxBounces

The equations behind it

What projectile motion is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

Kinematics

$v = \frac{Δ s}{Δ t}$
Average velocity
$a = \frac{Δ v}{Δ t}$
Average acceleration
$v = u + a t$
Velocity after a time — Constant acceleration only.
$s = u t + \frac{1}{2} a t^{2}$
Displacement from a time
$v^{2} = u^{2} + 2 a s$
Velocity from a displacement
$s = \frac{( u + v )}{2} t$
Displacement from the average velocity

Work, energy and power

$W = F s cos θ$
Work done by a force
$E_{k} = \frac{1}{2} m v^{2} = \frac{p ^{2}}{2 m}$
Kinetic energy
$Δ E_{p} = m g Δ h$
Change in gravitational potential energy — Near the surface, where g is constant.
$E_{elastic} = \frac{1}{2} k Δ x^{2}$
Elastic potential energy
$P = \frac{Δ W}{Δ t} = F v$
Power
$η = \frac{E _{useful}}{E _{input}}$
Efficiency