About the Momentum Calculator
Momentum measures how hard it is to stop a moving object. A slow truck and a fast baseball can both be dangerous because each carries a lot of momentum, and the force needed to stop them depends on how quickly that momentum is removed.
Momentum matters most in collisions. Whenever objects interact only with each other, their total momentum before and after is exactly the same. That single law lets you predict the outcome of crashes, recoils and explosions without knowing the complicated forces during the impact.
What is momentum?
Linear momentum is mass times velocity, p = mv, measured in kilogram-metres per second. Because velocity has a direction, momentum is a vector. In a straight-line problem you choose one direction as positive, and velocities and momenta pointing the other way are negative.
Impulse is the change in momentum produced by a force acting over a time interval, J = FΔt = Δp. The same impulse can come from a large force for a short time or a small force for a longer time, which is the idea behind padding, airbags and bending your knees when you land.
How to use the Momentum Calculator
- Select momentum, impulse or 1D collision.
- For p = mv or J = FΔt, enter two values and leave one empty.
- For a collision, choose elastic or perfectly inelastic.
- Enter m₁, u₁, m₂ and u₂, using negative signs for leftward motion.
- Press Calculate and read the final velocities and energy check.
Formulas
Momentum
Impulse–momentum theorem
Conservation of momentum
Perfectly inelastic
Elastic (1D)
Worked example: Momentum of a 1000 kg car at 20 m/s
A 2 kg cart moving at 3 m/s hits a 1 kg cart at rest and they lock together. Total momentum before is kg·m/s, so afterwards m/s.
Kinetic energy drops from J to J. The missing 3 J, a third of the original, becomes heat, sound and deformation of the couplings.
The solution below is generated by the same engine that powers the calculator above.
Final answer
momentum p = 20000 kg·m/s
How we read your problem
Use the momentum formula, rearranged for the unknown, in SI units.
Step-by-step solution
Formula: momentum
Substitute the known values
Formulas used
momentum
Momentum is a vector with the same direction as the velocity; in one dimension the sign shows the direction. A 1000 kg car at 20 m/s and a 10 kg ball at 2000 m/s carry the same momentum.
Verification
- Substituted every value back into the formula: both sides agree ✓
Detailed explanation
Elastic and inelastic collisions
Every isolated collision conserves momentum, but kinetic energy is a different story. In an elastic collision, such as between billiard balls or gas molecules, kinetic energy is conserved too. In an inelastic collision some kinetic energy is converted to other forms, and in a perfectly inelastic one the objects stick and move off together.
Real collisions usually fall in between. The two cases in this calculator are the limits: the elastic result gives the largest possible rebound, and the perfectly inelastic result gives the greatest possible loss of kinetic energy consistent with conservation of momentum.
Equal masses and Newton's cradle
When two equal masses collide elastically and one is at rest, the formulas give v₁ = 0 and v₂ = u₁: the moving ball stops dead and the other leaves with its full speed. That is exactly what happens in a Newton's cradle and in a good snooker stun shot.
If a light object hits a much heavier one at rest, it bounces back with almost its original speed while the heavy object barely moves. A heavy object hitting a light one keeps going at nearly the same speed and the light one shoots off at almost twice that speed.
Impulse and stopping safely
A 0.145 kg baseball arriving at 40 m/s and leaving at 40 m/s in the opposite direction changes its momentum by 11.6 kg·m/s. If the bat is in contact for 0.7 ms, the average force is over 16,000 N.
Crumple zones and airbags work the other way: they increase Δt. Stopping a 70 kg passenger from 15 m/s means an impulse of 1050 N·s; spread over 0.1 s instead of 0.01 s, the average force falls from 105,000 N to 10,500 N.
Common mistakes
- Ignoring the sign of a velocity in the opposite direction.
- Assuming kinetic energy is conserved in every collision.
- Adding speeds instead of momenta when objects stick together.
- Forgetting that impulse is a change in momentum, not momentum itself.
- Mixing grams and kilograms for the two masses.
Tips
- Draw before-and-after sketches with arrows and a positive direction.
- Check that total momentum is the same before and after.
- Use the kinetic energy change to classify a collision.
- For explosions from rest, total momentum afterwards is zero.
- Use the work and energy calculator to follow the energy lost.
Frequently asked questions
How do you calculate momentum?
Multiply mass in kilograms by velocity in metres per second: p = mv. A 1000 kg car at 20 m/s has 20,000 kg·m/s.
Is momentum always conserved?
Total momentum is conserved whenever no net external force acts on the system, which is a very good approximation during brief collisions.
What is impulse?
Impulse is force multiplied by the time it acts, J = FΔt. It equals the change in momentum of the object.
What is the difference between elastic and inelastic collisions?
Both conserve momentum. Elastic collisions also conserve kinetic energy; inelastic ones convert some of it to heat, sound and deformation.
How do I find the velocity after objects stick together?
Divide the total momentum by the total mass: v = (m₁u₁ + m₂u₂)/(m₁ + m₂).
Why can a small object have large momentum?
Momentum depends on velocity as well as mass, so a light object moving very fast, such as a bullet, can carry significant momentum.
What are the units of momentum?
Kilogram-metres per second (kg·m/s), which are the same as newton-seconds (N·s), the unit of impulse.
Solve a momentum problem
Choose momentum, impulse or a collision above and enter your values.