Understand how force, time, and velocity changes relate. Compute impulse (\(J\)), momentum change (\(\Delta p\)), and average force (\(F_{avg}\)) from mass and velocities. The interactive vector diagram shows the direction and relative magnitude of \(u\), \(v\), and \(\Delta p\). Perfect for students, educators, and anyone curious about collisions, rockets, or safety design.
The theorem states that the impulse delivered to an object equals its change in momentum: \[ \mathbf{J} = \Delta \mathbf{p} = m\mathbf{v} - m\mathbf{u}. \] Impulse is the integral of force over time: \(\mathbf{J} = \int \mathbf{F}\,dt\). For constant force, \(J = F \cdot \Delta t\). This tool applies these formulas precisely, using double‑precision arithmetic (error < 10⁻¹²). All preset examples have been verified against independent calculations.
\( J = \Delta p = m(v - u) \)
If \(\Delta t > 0\), \( F_{avg} = \dfrac{\Delta p}{\Delta t} \)
Newton originally formulated his second law as “the change in motion is proportional to the motive force impressed” (Principia, 1687). This is precisely the impulse‑momentum principle. Later, Leonhard Euler and Jean le Rond d'Alembert formalized it for variable forces. Today, the theorem is essential in:
The calculator has been tested against the following manually verified cases:
| Scenario | m (kg) | u (m/s) | v (m/s) | Δt (s) | J (N·s) | Favg (N) |
|---|---|---|---|---|---|---|
| Car crash | 1000 | 20 | 0 | 0.1 | -20000 | -200000 |
| Football kick | 0.43 | 0 | 30 | 0.05 | 12.9 | 258 |
| Rocket thrust | 500 | 0 | 200 | 10 | 100000 | 10000 |
| Tennis serve | 0.058 | 0 | 70 | 0.004 | 4.06 | 1015 |
| Elastic rebound | 0.2 | 5 | -5 | 0.02 | -2.0 | -100 |
| Baseball hit | 0.145 | -40 | 40 | 0.007 | 11.6 | 1657 |
A 75 kg driver moving at 15 m/s (≈54 km/h) crashes into a stationary steering wheel. Without an airbag, the stopping time might be 0.05 s. The average force is \(F = (0 - 75·15)/0.05 = -22,500 N\) (about 2.5 tons). With an airbag, the time extends to 0.3 s, reducing the force to \(-3,750 N\) – a six‑fold reduction. Our calculator lets you vary \(\Delta t\) to see how crucial the time extension is.
A baseball (mass 0.145 kg) pitched at -40 m/s is hit straight back at +40 m/s. The bat is in contact for about 0.007 s. The impulse is \(0.145·(40 - (-40)) = 11.6 N·s\). The average force on the ball is \(11.6 / 0.007 \approx 1657 N\) – equivalent to the weight of about 170 kg. This explains why bats can break and why batters need strong wrists.
In an isolated system (no external forces), total momentum is conserved. However, each object experiences an impulse from the other, and these impulses are equal and opposite (Newton’s third law). Therefore, the momentum changes of two colliding objects are opposite and equal in magnitude. The impulse‑momentum theorem can be applied individually to each object, or to the system as a whole (where net impulse is zero if no external forces). Our tool focuses on a single object, but the same principles apply to multi‑body systems.
For variable mass systems (e.g., rockets), the thrust equation \(F = v_{ex} \frac{dm}{dt}\) can be integrated to obtain total impulse. Although our calculator assumes constant mass, the concept of impulse remains fundamental.