Momentum & Impulse Calculator

Calculate momentum, impulse, and change in velocity using the impulse-momentum theorem.

Negative = opposite direction
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Impulse Is the Time Accumulation of Force

Impulse changes momentum: the area under a force-versus-time curve equals the change in an object’s momentum. For a constant force, J=FΔt. More generally J=∫Fdt=Δp. Since momentum p=mv in the nonrelativistic regime, impulse can change the magnitude or direction of velocity depending on the force direction.

The same momentum change can be produced by a large force acting briefly or a smaller force acting for longer. This is why increasing collision time with padding, airbags, or flexible structures can reduce average force even when the required momentum change is similar.

p=mv,   J=Δp=∫Fdt,   constant force: J=FΔt
SymbolMeaningWhy it appears / units
pLinear momentumkg·m/s; vector quantity.
JImpulseN·s, equivalent to kg·m/s.
FNet forceN; vector whose time integral gives impulse.
ΔtInteraction times for constant or average-force estimates.

Momentum and impulse are directional. In one-dimensional problems, choose a positive direction and keep velocity and force signs consistent. For variable force, average force equals impulse divided by the time interval.

Impulse and momentum must carry the same units and direction convention. N·s equals kg·m/s. For a fixed momentum change, spreading the impulse over a longer time lowers the average force; this is the principle behind airbags, padding, and longer stopping times.

Worked Examples

Example 1: Cricket ball impact: m=0.156kg, v₁=40m/s, v₂=-30m/s, Δt=0.005s
Δp=0.156×(-70)=-10.9kg·m/s
Result: F_avg=10.9/0.005=2184 N
Forces in ball sports are enormous
Example 2: Airbag: extends Δt from 0.005s to 0.05s
Same Δp=m×Δv
Result: Force 10× smaller = major safety benefit
Airbags increase impact time to reduce peak force
Example 3: Ball reversal
m=0.20kg, vi=+10m/s, vf=−15m/s
Result: J=−5.0N·s
Reversing direction makes the momentum change larger than merely stopping.
Example 4: Average impact force
|J|=5N·s over 0.10s
Result: |Favg|=50N
Doubling the stopping time would halve the same average force for the same impulse.

Common Mistakes

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Using speed without direction

Momentum is a vector, so opposite velocities must have opposite signs in one-dimensional calculations.

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Using one force instead of net force for impulse

The change in total momentum equals net external impulse, not necessarily the impulse from one selected force.

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Assuming force is constant when it varies strongly

For variable force, use the area under the F-t graph or an average force consistent with the same total impulse.

Frequently Asked Questions

Why do airbags save lives?
J=FΔt=Δp is fixed (same collision). Increasing Δt (airbag) reduces F proportionally. Without airbag: Δt≈5ms, F≈2000N. With airbag: Δt≈50ms, F≈200N. 10× force reduction prevents fatal injury.
Momentum vs kinetic energy in collisions?
Momentum is always conserved in collisions. Kinetic energy is only conserved in elastic collisions. In inelastic collisions, KE converts to heat, sound, and deformation. Crumple zones deliberately convert KE to deformation.
Why do N·s and kg·m/s represent the same unit?
One newton is kg·m/s². Multiplying by seconds gives kg·m/s, exactly the SI unit of momentum.
Can impulse change direction without changing speed?
Yes. A force perpendicular to velocity can redirect momentum while leaving speed unchanged, as in an idealized curved path.
Why do airbags reduce injury force?
They increase the time and distance over which a passenger’s momentum changes. For a similar impulse, a longer interaction time lowers average force.
Does impulse conservation exist?
Impulse itself is not a conserved quantity. In an isolated interaction, internal impulses occur in equal and opposite pairs, leading to conservation of total momentum.
How does a force–time graph give impulse?
Impulse is the time integral of force, J=∫Fdt, so it equals the signed area under a force-versus-time graph. For constant force this becomes FΔt. The impulse equals the change in momentum, including direction, so negative graph area corresponds to a momentum change in the negative chosen direction.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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