Impulse and Momentum Calculator

Enter mass, initial velocity, final velocity, and the time interval to find momentum before and after, the impulse (J = Δp = mΔv), and the average force applied using the impulse-momentum theorem.

Quick Facts

Impulse-Momentum Theorem
J = Δp = m(v_f − v_i)
The net impulse on an object equals its change in momentum.
Impulse from Force
J = F × Δt
Average force multiplied by the time interval it acts over.
Shared Units
1 N·s = 1 kg·m/s
Impulse and momentum are measured in the same physical units.

Your Results

Calculated
Impulse / Momentum Change (J = Δp)
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N·s (equivalent to kg·m/s)
Average Force (F)
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F = J / Δt, in newtons
Initial Momentum (p_i)
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p_i = m × v_i, in kg·m/s
Final Momentum (p_f)
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p_f = m × v_f, in kg·m/s

Ready

Enter mass, initial and final velocity, and the time interval, then press Calculate.

Formula and Method for Impulse and Momentum

Momentum is a measure of an object's motion, defined as mass times velocity: p = mv. When a net force acts on an object over a time interval, it changes that object's momentum — this effect is called impulse, J = FΔt. The impulse-momentum theorem connects the two directly: the impulse delivered to an object equals its change in momentum, J = Δp = m(v_f − v_i). This calculator uses that theorem to find momentum before and after, the impulse, and the average force from your mass, velocity, and time inputs.

The impulse-momentum theorem

Enter the object's mass, its velocity before the interaction (v_i), its velocity after the interaction (v_f), and the time interval (Δt) over which the change happens. The calculator first finds the initial momentum (p_i = m × v_i) and final momentum (p_f = m × v_f), then subtracts them to get the change in momentum (Δp = p_f − p_i). By the impulse-momentum theorem, this change is exactly equal to the impulse delivered, J = Δp. Dividing that impulse by the time interval gives the average force applied, F = J / Δt — this comes directly from Newton's second law written as F = m(Δv/Δt) = Δp/Δt. A negative result simply means the impulse or force acts opposite to your chosen positive direction (for example, a ball bouncing back after a collision).

Getting accurate results

  • Pick one positive direction and stick to it — if an object reverses direction (like a ball bouncing off a wall), its velocity after the bounce should be entered as negative.
  • The force calculated here is the average force over the interval, not the peak force. Real impacts (bat hits, crashes, collisions) involve forces that spike well above the average during a very short Δt.
  • Keep units consistent: this calculator converts mass and velocity to kilograms and meters/second internally, and Δt must always be entered in seconds.
  • A shorter Δt for the same momentum change produces a much larger average force — this is why airbags, crumple zones, and padded gloves work by extending contact time.

Frequently Asked Questions

What is the difference between impulse and momentum?
Momentum (p = mv) describes the motion an object has at one instant. Impulse (J) is what changes that motion — it is the effect of a force acting over a time interval, J = FΔt. The impulse-momentum theorem states they are directly linked: the impulse applied to an object equals its change in momentum, J = Δp = m(v_f − v_i).
What units are used for impulse and momentum?
In SI units, momentum is measured in kilogram-meters per second (kg·m/s) and impulse is measured in newton-seconds (N·s). These are equivalent units — 1 N·s = 1 kg·m/s — which is why impulse and momentum change always match numerically.
How do I find the average force from impulse?
Rearrange the impulse formula J = FΔt to solve for force: F = J / Δt, where J is the impulse (equal to the momentum change m·Δv) and Δt is the duration the force acts over. Shortening the contact time increases the average force needed to produce the same momentum change, which is why padding, airbags, and follow-through technique matter in collisions.
Why does a shorter collision time increase the force?
Because impulse (Δp) is fixed by how much the velocity changes, and J = FΔt, force and time are inversely related for a given impulse. A car crashing into a rigid wall stops in a very short Δt, producing a huge average force; crumple zones and airbags extend Δt so the same momentum change happens with much less force on the occupants.

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