Impulse-Momentum Theorem Calculator
Solve FΔt = mΔv for force, time interval, mass, or velocity change. Enter any three values and get the fourth with step-by-step working shown.
The impulse–momentum theorem
Impulse — force applied over time — equals the resulting change in momentum:
- F — force, in newtons (N)
- Δt — time interval over which the force acts, in seconds (s) — always positive
- m — mass, in kilograms (kg) — always positive
- Δv — change in velocity, in metres per second (m/s) — can be negative
This follows directly from Newton's second law: since a = Δv/Δt, substituting into F = ma gives F = mΔv/Δt, which rearranges to FΔt = mΔv.
Enter any three values and leave the fourth blank to solve for it.
Worked example
A 2 kg object speeds up from 0 to 5 m/s over 4 seconds. What average force acted on it?
- Formula: FΔt = mΔv
- Substitute: F × 4 = 2 × 5
- Result: F = 10 / 4 = 2.5 N
Solving for the other variables
- Mass: m = FΔt / Δv — enter F, Δt, and Δv, leave m blank.
- Velocity change: Δv = FΔt / m — enter F, Δt, and m, leave Δv blank.
- Time interval: Δt = mΔv / F — enter m, Δv, and F, leave Δt blank.
Common mix-ups
- Confusing Δv with v. Δv is the change in velocity (final minus initial), not the final velocity itself.
- Sign errors on Δv. If an object slows down or reverses, Δv is negative — carry that sign through to get the correct signed force.
- Forgetting Δt must be positive. A zero or negative time interval isn't physically meaningful.
Where this shows up
The impulse–momentum theorem is the standard tool for analyzing collisions, impacts, and any short-duration force event where you know mass and measured velocity change but not the force curve itself — sports impacts, vehicle crash tests, and rocket staging. See impulse for the simpler J = Ft form and momentum for p = mv.
Frequently asked questions
- What is the impulse–momentum theorem?
- It states that impulse equals the change in momentum: FΔt = mΔv, where F is force (N), Δt is the time interval (s), m is mass (kg), and Δv is the change in velocity (m/s).
- Where does the impulse–momentum theorem come from?
- It follows directly from Newton's second law. Since a = Δv/Δt, substituting into F = ma gives F = mΔv/Δt, and multiplying both sides by Δt gives FΔt = mΔv.
- How do I find the change in velocity if I know force, time, and mass?
- Rearrange to Δv = FΔt / m. Enter force, time interval, and mass, and leave the velocity-change field blank to have it solved automatically.
- Can the velocity change be negative?
- Yes. A negative Δv means the object slowed down or reversed direction. Time interval and mass must stay positive, but force and velocity change can be either sign.
- How is this different from the impulse calculator (J = Ft)?
- J = Ft calculates impulse from force and time alone. This calculator uses the equivalent statement that impulse also equals mΔv, letting you solve for mass or velocity change directly instead of impulse itself.
- What is a typical use case for this formula?
- It's used to find the average force during a collision or impact when you know the mass and how much its velocity changed over a measured time — for example, a ball's velocity change on being hit by a bat over the brief contact time.