Impulse and Momentum Calculator
Use this calculator to solve any variable in the impulse-momentum theorem: impulse (J), force (F), contact time (t), mass (m), initial velocity (v1), or final velocity (v2). Choose a solve mode, fill in the known values, and the missing quantity appears instantly with a full worked solution. Unit switches let you work in metric or imperial throughout.
Formula
Worked example
A 0.5 kg ball rolling at 0 m/s is struck so it leaves at 10 m/s. p1 = 0.5 x 0 = 0 kg·m/s. p2 = 0.5 x 10 = 5 kg·m/s. Impulse = 5 - 0 = 5 N·s. If contact lasted 0.02 s, the average force was 5 / 0.02 = 250 N.
What is impulse in physics?
Impulse (J) is the product of average force and the time interval over which that force acts: J = F x t. It measures the total "push" delivered to an object rather than the instantaneous force alone. A small force applied for a long time can deliver the same impulse as a large force applied briefly. Impulse is a vector quantity measured in newton-seconds (N·s) or equivalently kilogram-metres-per-second (kg·m/s).
What is momentum and how does it relate to impulse?
Momentum (p) is the product of an object's mass and velocity: p = m x v. It is a vector quantity measured in kg·m/s. The impulse-momentum theorem states that the impulse delivered to an object equals its change in momentum: J = delta_p = p2 - p1 = m x (v2 - v1). This single equation links force, time, mass, and velocity change, which is why the two concepts are almost always taught together. If you know any four of the five quantities (J, F, t, m, and dv), you can solve for the fifth.
How to use the nine solve modes
Select the quantity you want to find from the "Solve for" menu. The calculator then unlocks the relevant input fields. For example, to find the average force during a cricket ball catch, select "Force (F)", enter the ball's mass and initial and final velocities, and specify the contact time. To find how long a car airbag must inflate to limit the peak deceleration force on a passenger, select "Contact time (t)". Every mode shows the full step-by-step derivation in the worked solution panel so you can see exactly how the answer is reached.
Why contact time matters: crumple zones, airbags, and padding
The impulse-momentum theorem has a critical practical implication: for a fixed change in momentum (such as stopping a moving vehicle or a falling person), a longer contact time means a smaller average force. This is the engineering principle behind car crumple zones, which extend the collision duration from milliseconds to tens of milliseconds, cutting peak forces on occupants by an order of magnitude. Airbags extend the time the head decelerates; gym mats, boxing gloves, and cricket pads work on the same principle. Shorter contact time concentrates the same impulse into a larger peak force, which is why a karate chop is more damaging than a slow push.
Conservation of momentum and collisions
In a closed system with no external forces, total momentum is conserved: the sum of momenta before a collision equals the sum after. Impulse analysis is the bridge between individual object changes and the overall system: the impulse one object exerts on another is equal and opposite (Newton's third law), so the two changes in momentum cancel and total momentum is unchanged. This calculator handles the single-object impulse calculation; for two-body collisions, pair it with a conservation-of-momentum calculator.
Common impulse and momentum examples
| Scenario | Mass | Velocity change | Approximate impulse |
|---|---|---|---|
| Cricket ball catch (hard pace) | 0.16 kg | 40 m/s to 0 | 6.4 N·s |
| Football header | 0.43 kg | 20 m/s to -5 m/s | 10.8 N·s |
| Rifle bullet (7.62 mm) | 0.008 kg | 880 m/s to 0 | 7.0 N·s |
| Car airbag (occupant head) | 6 kg | 15 m/s to 0 | 90 N·s |
| Tennis serve return | 0.058 kg | 70 m/s to -40 m/s | 6.4 N·s |
| Golf drive (club on ball) | 0.046 kg | 0 to 75 m/s | 3.5 N·s |
Approximate values for familiar real-world scenarios.
Frequently asked questions
What are the units of impulse?
Impulse is measured in newton-seconds (N·s). Because 1 N = 1 kg·m/s², multiplying by seconds gives 1 N·s = 1 kg·m/s, which is the same unit as momentum. The two are physically the same quantity, just arriving from different directions of the same equation.
What is the impulse-momentum theorem?
The impulse-momentum theorem states that the impulse acting on an object equals the change in that object's momentum: J = F x t = m x (v2 - v1) = p2 - p1. It follows directly from Newton's second law (F = m x a) by integrating over time. The theorem is useful when force is applied for a finite duration and you want to know the resulting velocity change without needing to know the detailed shape of the force-time curve.
How do I calculate impulse from force and time?
Multiply the average applied force (in newtons) by the contact time (in seconds): J = F x t. For example, a force of 200 N applied for 0.05 s delivers an impulse of 200 x 0.05 = 10 N·s. If the force varies over time, the impulse equals the area under the force-time graph.
Is impulse the same as momentum?
They have the same units and the impulse equals the change in momentum, but they are not the same thing. Momentum is the current state of an object's motion (p = m x v), while impulse is the change in that state caused by a force (J = delta_p). Applying an impulse shifts the object from one momentum to another.
What is a negative impulse?
A negative impulse means the force acted in the opposite direction to the positive axis you defined. For example, if you define the positive direction as "forward," a braking force produces a negative impulse and reduces the forward momentum. The magnitude of the impulse is the same; the sign tells you the direction of the force and whether momentum increased or decreased.
Why is contact time important in sports and safety engineering?
Because the impulse (momentum change) is fixed by the initial and final velocities, a longer contact time means a lower average force. A cricket fielder who "gives" with a catch spreads the ball's momentum change over more time, reducing the peak force on the hands. Crumple zones, airbags, and padded surfaces all extend contact time to reduce peak force for the same impulse.