Impulse and Momentum Calculator
Find the change in velocity from a force and contact time, or the average force needed to change an object's velocity, using the impulse-momentum theorem.
💥 What is the Impulse and Momentum Calculator?
Impulse is the product of a force and the time it acts, and momentum is mass times velocity. The impulse-momentum theorem links them: the impulse delivered to an object equals the change in its momentum, J = Favg Δt = Δp = m (vf − vi). This calculator applies that theorem in two directions. In the first mode you know the push (force and contact time) and it returns the change in velocity and the final velocity. In the second mode you know how the velocity changed and how long the change took, and it returns the average force that must have acted.
The theorem is the tool of choice for any short, hard interaction where the force is huge but brief and hard to measure directly: a bat hitting a ball, a golf club striking a ball, a car crash, a person landing from a jump, a catch, a kick or a hammer blow. Engineers use it to size airbags and crumple zones, coaches use it to explain follow-through, and students use it to solve collision problems without needing the detailed force curve. You only need the masses, the velocities before and after, and an estimate of the contact time.
Two points are commonly misunderstood. First, impulse is not the same as momentum: momentum describes a moving object, impulse describes what a force does to it over time. Second, the force the theorem gives is an average over the contact time. A real impact force rises quickly, peaks and falls, so the peak can be well above the average. The second mode estimates the peak for two idealised pulse shapes so you can see how much headroom a design needs. Impulse here also has nothing to do with specific impulse in rocketry, which measures engine efficiency in seconds, although both ideas start from force acting over time.
The chart shows the practical lesson of the theorem. For a fixed change in momentum the average force is inversely proportional to the contact time, so a curve of force against time is a hyperbola. Doubling the stopping time halves the force. If you want the forces and energies separately, see the force calculator, the kinetic energy calculator and the velocity calculator.
📐 Impulse-Momentum Formulas
The theorem follows from Newton's second law, F = dp/dt, by integrating over the contact time. For a force that varies with time the impulse is the area under the force-time curve, and Favg is that area divided by Δt. The half-sine factor π/2 comes from the fact that the average of sin over half a cycle is 2/π, so peak divided by average is π/2, about 1.571. The triangular factor 2 follows because a triangle's area is half its base times its height. Real impacts rarely match either shape, so the peak factors are rules of thumb. Kinetic energy uses KE = ½ m v².
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Baseball struck by a bat
0.145 kg ball arriving at 40 m/s and leaving at 45 m/s the other way, contact 0.7 ms (illustrative values)
Example 2 — Car stopping with and without a crumple zone
1,200 kg car at 15 m/s stopped in 50 ms (rigid) versus 150 ms (crumple zone)
Example 3 — Catching a ball with soft hands versus stiff hands
0.15 kg ball caught at 25 m/s, stopped in 20 ms (stiff) versus 200 ms (soft)
Example 4 — Golf club striking a ball
0.0459 kg ball leaving the club at 70 m/s after a 0.5 ms contact (illustrative values)
Example 5 — Pushing a stalled car (velocity mode)
1,200 kg car at rest pushed with a steady 5,000 N for 4 s (friction ignored)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the impulse-momentum theorem?
The impulse-momentum theorem says the impulse on an object equals its change in momentum: J = F_avg x delta t = delta p = m (v_f - v_i). A larger force or a longer contact time gives a larger impulse and therefore a larger change in velocity. It is Newton's second law written in terms of momentum, and it holds even when the force changes during the contact.
How do you calculate impulse?
Multiply the average force by the time it acts: J = F x delta t. A 5,000 N force acting for 4 s gives J = 20,000 N s. Equivalently, impulse is mass times the change in velocity, m (v_f - v_i), so both routes must give the same number. The unit is the newton second, which equals one kilogram metre per second.
How do you find average impact force from velocity and time?
Use F_avg = m (v_f - v_i) / delta t. A 1,200 kg car going 15 m/s that stops in 0.05 s needs F = 1,200 x 15 / 0.05 = 360,000 N. Pick the final velocity carefully: zero if the object stops, a negative value if it rebounds. The sign of F tells you the direction of the push.
Why does a longer contact time reduce the force?
For a fixed change in momentum, F_avg = delta p / delta t. The numerator is set by the masses and velocities, so the only way to lower the force is to lengthen the time. Doubling the stopping time halves the average force. This is why crumple zones, airbags, padded floors and bent knees reduce injury.
What is the difference between impulse and momentum?
Momentum p = m v is a property of a moving object at one instant. Impulse is the effect of a force acting over a time, and it equals the change in momentum between two instants. Both are measured in N s (kg m/s), but momentum describes the state while impulse describes the change.
What is the difference between average force and peak force?
The average force is the impulse divided by the contact time. The peak force is the largest instantaneous value. For a symmetric triangular pulse the peak is 2 times the average, and for a half-sine pulse it is pi/2, about 1.57 times the average. Real impacts have irregular shapes, so treat these as rules of thumb.
What is g-force and how is it calculated here?
G-force in this calculator is the average force divided by the object's weight, F / (m g), with g = 9.80665 m/s squared. A result of 10 g means the average force is ten times the weight of the object. It is a convenient way to compare very different masses, but it is not the same as the acceleration felt by every part of a body.
Is this the same as specific impulse in rocketry?
No. Impulse here is force times time, in N s. Specific impulse in rocketry is a measure of engine efficiency, thrust per unit weight flow of propellant, expressed in seconds. The two share a name because total engine impulse is thrust times burn time, but this calculator solves collisions and pushes, not rocket engine efficiency.