Momentum Calculator
Momentum is a vector, and this page keeps it one. Enter a mass and a signed velocity and it returns p = m × v with the sign attached: 0.145 kg at +40 m/s is 5.8 kg m/s, and 2 kg at -3 m/s is -6 kg m/s. Add a final velocity and the Change cell gives the impulse, m(v_final - v_initial), so the same 0.145 kg leaving at -35 m/s changes by -10.875 kg m/s, and a note says in words that subtracting the two sizes would have given -0.725 kg m/s instead, because that is the slip this calculation invites. It runs the other way too, solving velocity from a momentum you already know, so 5.8 kg m/s over 0.145 kg comes back as +40 m/s. Mass takes kilograms, grams, milligrams, tonnes, pounds or ounces, velocity takes m/s, km/h, cm/s, km/s, ft/s, mph or knots, and the arithmetic runs in SI with the conversion printed whenever a unit is not the SI one.
Mass times velocity, with the sign of the velocity carried through: a velocity of -3 m/s is a momentum that points the other way. Add a final velocity and the change in momentum comes with it.
and . Momentum carries the dimensions , so its SI unit is the kilogram metre per second, which is the newton second.
The direction reverses between the two velocities, so the change is not the difference of the two sizes: subtracting those gives -0.725 kg m/s, while the signed change is -10.875 kg m/s. Momentum is a vector, and on one axis the sign is the direction.
Three cases, worked
A signed velocity, an impulse, and a velocity solved back out of a momentum. Every line below is computed by the code that answers the calculator above, so the examples and the tool can never disagree.
Common questions
- How do you calculate momentum?
- Multiply the mass by the velocity, keeping the sign of the velocity: p = m × v. A 0.145 kg baseball at +40 m/s carries 5.8 kg m/s, and a 2 kg trolley at -3 m/s carries -6 kg m/s. The line of arithmetic is printed under the answer with your own figures in it, so a number typed into the wrong field is easy to spot.
- Can momentum be negative?
- Yes, and the sign is the whole point. The tool works on one axis, so a velocity entered as negative points the other way and its momentum carries the same sign: 2 kg at -3 m/s is -6 kg m/s. Mass itself is a size rather than a direction, so a negative mass is refused with a message saying the direction belongs in the velocity.
- What is the change in momentum, and why is it not the difference of the two speeds?
- The change is m(v_final - v_initial), which is the same as p_final - p_initial, and it is the impulse delivered. The two ways of working it out part company the moment the direction reverses: a 0.145 kg ball arriving at +40 m/s and leaving at -35 m/s changes by -10.875 kg m/s, while subtracting the two sizes gives -0.725 kg m/s. The signed figure sits in the Change cell and the other one is named in a note, so they are never mistaken for each other.
- How do I find velocity from momentum?
- Set Solve for to Find velocity, then enter the momentum and the mass: v = p ÷ m. A momentum of 5.8 kg m/s on 0.145 kg comes back as +40 m/s, and a negative momentum returns a velocity pointing the other way. A mass of zero is refused with the reason rather than returned as an infinity.
- What is the SI unit of momentum and its dimensional formula?
- The kilogram metre per second, kg m/s, which is the newton second, N s. Its dimensional formula is M L T⁻¹: one of mass, one of length and one over time. That line sits under the readout, and the momentum can be shown in kg m/s, N s, g cm/s or lb ft/s.
- Does it work in pounds and miles per hour?
- Yes. Mass takes kilograms, grams, milligrams, tonnes, pounds or ounces, and velocity takes metres per second, kilometres per hour, centimetres per second, kilometres per second, feet per second, miles per hour or knots. Every factor is a definition rather than a measurement: a pound is 0.45359237 kg and a mile per hour is 0.44704 m/s. The arithmetic runs in SI, and a note shows the converted figures so you can check them.
- Is this relativistic momentum?
- No, it is the classical p = m × v, and the number is never quietly corrected. What the page does instead is measure the gap: from a hundredth of the speed of light upward a note names the shortfall, and at a tenth of light speed it reads that the classical value falls 0.501256289338% short of the relativistic γmv. Past light speed the note says that nothing carrying mass reaches that, and that the figures are the classical formula applied as typed.
- What does the change look like when nothing reverses?
- It is the same arithmetic with the sign left alone. A 1200 kg car going from 30 m/s to 10 m/s changes by -24000 kg m/s, negative because the momentum fell while the direction held, and no reversal note appears. The figure is the change in momentum and nothing else: it describes no collision and no force.
- How precise are the results?
- The arithmetic is ordinary double precision, the same as a spreadsheet, and figures are shown to twelve significant figures, with whole numbers shown in full and very large or very small values in exponent form. Nothing is rounded before the answer is worked out. A product that runs past what a double can hold, or one so small it cannot be told from zero, is refused by name rather than shown as an infinity or a zero.
- Are my numbers stored or sent anywhere?
- They stay in this browser. The last set you typed is remembered locally so the page opens where you left it, and nothing is uploaded. If storage is blocked or full the calculator still works, it simply forgets.
Exact arithmetic on one signed axis: momentum is mass times velocity, and the change in momentum is the impulse delivered. It is the classical formula, so it holds well below the speed of light and nowhere near it. It describes a number and says nothing about an impact, a collision or whether anything is safe.