Momentum & Impulse Calculator

How to use this calculator

Pick a positive direction and enter velocities with signs: an object that reverses direction has velocities of opposite sign, which is why rebounds deliver so much impulse. Enter the mass and both velocities to get the momenta and the impulse; add the contact time to get the average force.

The calculation also runs backwards. If a sensor measured the peak of a force plate reading, enter the average force, contact time and mass to find the change in velocity. Or enter impulse and force to see how much longer an airbag or crumple zone must extend the collision to halve the force.

The formula

p = m v

J = Δp = p₂ − p₁ = m (v₂ − v₁)

J = F Δt

m: mass
v₁: initial velocity
v₂: final velocity
p₁: initial momentum
p₂: final momentum
J: impulse (change in momentum)
Δt: contact time
F: average force

Worked example

A 145 g baseball arrives at 40 m/s and leaves the bat at 50 m/s in the opposite direction. The bat is in contact for 0.7 ms.

Take the outgoing direction as positive, so v₁ = −40 m/s and v₂ = +50 m/s.
p₁ = 0.145 × −40 = −5.8 kg·m/s; p₂ = 0.145 × 50 = 7.25 kg·m/s; impulse J = p₂ − p₁ = 13.05 N·s.
Average force F = J / Δt = 13.05 / 0.0007 = 18,600 N.

The bat pushes on the ball with an average of about 18.6 kN — roughly 4,200 lbf — for less than a millisecond.

Assumptions and limits

Mass is constant during the interaction (no fuel burn or fragmentation).
F is the average force over Δt; the peak force is typically 1.5–2 times higher for a short impact.
One-dimensional motion: velocities are components along a single line.

Frequently asked questions

Is impulse the same as change in momentum?

Yes. The impulse–momentum theorem states J = Δp, so impulse has the same units as momentum: N·s = kg·m/s.

Why do airbags reduce injuries?

They do not change the impulse (you still go from your speed to zero), but they stretch the stopping time, so the average force F = J / Δt is much smaller.

What happens to momentum in a collision?

The total momentum of all colliding objects is conserved if no external force acts. Each object’s momentum changes by an equal and opposite impulse.