Change in Momentum Calculator

What a change in momentum means

Momentum is mass in motion: p = m x v. A change in momentum, written delta p, is what happens when a push, a pull or a collision changes how that mass moves. It answers a different question than momentum itself. Momentum says how hard an object is to stop. The change says how much the motion shifted.

The definition is a subtraction. Take the final momentum and subtract the initial momentum, keeping the sign, because momentum is a vector and direction is part of its value. A trolley slowing from 5 m/s to 2 m/s lost 3 m/s of speed, so a 4 kg trolley lost 12 kg m/s of momentum. A ball that bounces back changed even more, because its direction flipped and the speeds add.

That is why the change matters more than the raw number in collision problems. Two objects can carry the same momentum and behave nothing alike when they hit something, depending on how quickly their motion is brought to rest. The size of the momentum change is fixed by the speeds at each end. How much force it takes depends on a third number, the contact time, which is where the impulse connection comes in later in this page.

The formula, in two forms

The calculator starts from one line and rearranges it two ways:

delta p = m(v_final - v_original) = F x delta t

The left form comes straight from the definition of momentum, since subtracting m x v_original from m x v_final factors into m times the velocity difference. The right form is the impulse-momentum theorem, a rearrangement of Newton's second law: net force equals the rate of change of momentum, so force held for a time equals the momentum change it produces. The OpenStax physics text presents both forms of the theorem side by side.

Signs carry the direction. Set positive as the direction the object first moves. If the object speeds up, v_final is larger and delta p is positive. If it slows down, delta p is negative. If it reverses, v_final is negative and the change is bigger than either momentum alone, because the two speeds add instead of subtract.

Worked examples from the calculator

Each row below was run through the widget on this page and confirmed by an automated test:

Read the first row carefully. The baseball carries 5.8 kg m/s of momentum before the catch and none after, so the change is -5.8 kg m/s. Caught stiffly in a thousandth of a second, that needs an average force of 5800 N, close to the weight of half a ton, which is why a barehanded line drive stings. Carried back with the hands over a longer time, the same catch happens at a small fraction of that force.

The last two rows show the same momentum change, 12 kg m/s, built from different objects and times. A light ball reversing direction needs a hard, brief shove. A loaded trolley easing down needs a gentle one. The momentum bookkeeping is identical in both cases; only the contact time differs.

The second row is the standard stopping-distance problem in disguise. A 70 kg rider moving at 10 m/s carries 700 kg m/s. Stopping over 0.1 s puts the average force at 7000 N. Stretch the same stop to a full second, the way an air bag and crumple zone try to, and the force drops to 700 N.

A change of direction counts too

Speed is a number; velocity is a number plus a direction. A change in either one changes momentum. That distinction turns simple-looking problems into larger answers than people expect.

Take the 2 kg ball from the table, thrown at +3 m/s against a wall and bouncing back at -3 m/s. The speeds match, so it is tempting to call the change zero. But the momenta are +6 kg m/s and -6 kg m/s, and the change is -12 kg m/s, twice what a full stop would give. The Oregon State physics course notes stress this point: a change in momentum can be a change in direction, a change in size, or both.

The calculator handles this by treating velocity as a signed number. Enter the rebound speed with a minus sign and the subtraction does the rest. Enter both velocities as positive and the tool reads the result as a speed-up or slow-down along one line.

One caution belongs with this. Momentum is a vector in three dimensions, and this one-dimensional tool adds and subtracts along a single axis. A ball that strikes a wall at an angle carries two momentum components, and only the component along each axis can be entered here. For angled collisions, split the velocities into components first, then run each axis through the calculator.

Contact time changes the force, never the momentum change

The momentum change in a collision is fixed by the masses and speeds at each end. The force is not. Divide the same delta p by a different contact time and you get a different average force, which is the whole idea behind padding, helmets and crumple zones.

Five hundred times more contact time turns a bone-breaking force into a light one, and the momentum arithmetic never changed. The Physics Classroom makes the same point with a football example: an 800 N force held for 0.9 s gives an impulse of 720 N s, equal to a momentum change of 720 kg m/s, and 800 x 0.9 = 720.

A textbook example makes the stakes concrete. The Michigan State open physics text works a bicycle crash: a head with an effective mass of 5.0 kg hitting pavement at 6.0 m/s carries a momentum change of 30 kg m/s. Stopping in 2.0 ms needs 15000 N. A helmet liner stretching the stop to 10 ms cuts the average force to 3000 N, one fifth of it, for the identical momentum change.

This is also why the calculator separates the two outputs. The momentum change is physics you cannot negotiate; the force is engineering you can.

Units: kg m/s and N s are the same thing

Check the units on both sides of the impulse form. A newton is a kilogram meter per second squared by definition, so a newton-second works out to kg m/s. The two names describe one unit, which is why an impulse in N s and a momentum change in kg m/s can be set equal without a conversion factor.

Other unit pairs appear in older problems and British sources. One N s equals 1 kg m/s, 0.2248 pound-second of impulse, or 0.7376 pound-foot-second. Working from grams and centimeters, a g cm/s is 0.00001 kg m/s. The calculator keeps to kilograms, meters and seconds and prints both names for the result, so a figure like -5.8 reads as -5.8 kg m/s and, at the same time, -5.8 N s.

The one trap worth naming is pounds. Pounds-force and pounds-mass are different things, and a mass typed as a weight in pounds must be divided by 2.2046 to become kilograms before the formula works. A 145 pound catcher has a mass of about 65.8 kg, not 145 of anything.

Where the number shows up

Vehicle safety is the largest user of this arithmetic. Crash standards are built around stopping a car's momentum over as much time and distance as possible. Crumple zones fold so that the stop stretches from a few milliseconds to a tenth of a second or more. A 1500 kg car at 20 m/s carries 30000 kg m/s, and the difference between absorbing that in 0.05 s and in 0.5 s is the difference between an average force of 600000 N and 60000 N on the structure.

Sport follows the same logic from the other side. Catching instruction tells players to pull the hands back with the ball, adding contact time. Gymnasts bend the knees on landing. Boxers ride a punch by moving the head back with the blow. Each trick buys time, and time is the only thing that lowers the force for a momentum change that is already fixed.

Rocketry closes the loop in the opposite direction. A rocket engine throws mass out the back, and the momentum gained by the craft equals the momentum carried away by the exhaust, flipped in sign. The same delta p formula, applied twice with opposite signs, is why thrust ratings are quoted in newtons and why a small force held for minutes can move a spacecraft as much as a big one held for seconds.

In each of these settings the number that matters is the momentum change, because that is what the physics fixes. Everything after it, the force, the damage, the comfort of the ride, depends on how much time the change is given.

What the calculator returns

The calculator takes four inputs: mass in kilograms, the original velocity in m/s, the final velocity in m/s, and an optional contact time in seconds. It returns the initial momentum, the final momentum, the change between them with its sign, the same number named as an impulse, and, when the contact time is larger than zero, the average force that produced it. The direction row tells you whether the object sped up, slowed down, reversed or did not change.

Missing or zero inputs are handled without errors. A mass of 0 gives a momentum change of 0, since nothing is moving to change. A contact time of 0 leaves the force line out rather than dividing by zero, and the momentum results still print. Negative velocities are welcome and are the normal way to enter a rebound.

Everything is computed in your browser with plain JavaScript, and nothing you type leaves the page. Results are rounded to a sensible number of digits for reading, while the arithmetic behind them uses full precision.

The one limit to remember is the single axis. This tool adds and subtracts velocity along one line, which covers catches, crashes and bounces straight back. For glancing blows or angled collisions, resolve the velocities into components and run each axis separately, then combine the two momentum changes if you need the total.