INTERACTIONS IN ONE DIMENSION

Collision and Momentum Lab

Two carts. Impact, friction and momentum. Predict what happens next.

Horizontal track

x → right · no walls · view follows both carts · fixed-size symbols

Ready

Vectors Pause at impact Prediction mode

Solid arrows: velocity (m/s) · dashed arrows: momentum (kg·m/s). Separate scales; neither is force.

Speed

VELOCITY · CART 1

VELOCITY · CART 2

TIME

Motion over time

● Cart 1 · ● Cart 2 · dashed: total

Track momentum and energy.

● Kinetic energy● Dissipated energy

Orange includes friction heat and collision energy lost to heat, sound or deformation.

Immediately before and after impact
Quantity	Before	After

TRY A CHALLENGE

Bring cart 1 to a stop.

This challenge uses zero friction. Keep m₁ = 2 kg and u₁ = 3 m/s. Choose m₂ and u₂ so cart 1 stops immediately after an elastic collision.

Run a collision to test your solution.

Show a hint

For e = 1 and v₁ = 0: (m₁ − m₂)u₁ + 2m₂u₂ = 0. Make sure the carts approach.

More experiments

Connect the carts at a target velocity of +1 m/s.
Have a partner hide m₂; infer it from the measured velocities.
With cart 2 at rest, vary its mass to maximize its gained kinetic energy.
Recreate a partner’s run from their momentum graph.
Predict both final velocities before every run.

The physics, unpacked

u₁ and u₂ below mean velocities immediately before impact, after any frictional slowing.

p = mv · K = ½mv²
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
v₂ − v₁ = e(u₁ − u₂)
J₁ = m₁(v₁ − u₁) = −J₂
V_CM = (m₁u₁ + m₂u₂)/(m₁ + m₂)

Between impacts: F = −μmg sign(v), a = −μg sign(v). Friction stops each cart at v = 0 without reversing it. COM velocity can change under friction; momentum is conserved across the instantaneous collision.

Final velocity equations

v₁ = [m₁u₁ + m₂u₂ − m₂e(u₁ − u₂)]/(m₁ + m₂)

v₂ = [m₁u₁ + m₂u₂ + m₁e(u₁ − u₂)]/(m₁ + m₂)

Impulse J is a change in momentum (N·s). Force is a rate of momentum change (N). Impact is instantaneous here, so there is no finite peak force to display. At e = 0 the carts stay connected.