Free Momentum Calculator
Calculate linear momentum p = m·v from an object's mass and velocity, with an optional impulse (J = F·Δt) section. Results update instantly in SI units (kg·m/s).
Enter mass and velocity to find linear momentum p = m·v. SI units: kilograms (kg) and m/s, giving momentum in kg·m/s.
Optional — impulse J = F·Δt. By the impulse–momentum theorem this equals the change in momentum (Δp), in N·s = kg·m/s.
p = m · v is a vector pointing in the direction of motion; a negative value means the opposite direction. Momentum (kg·m/s) differs from kinetic energy ½mv² (joules). Impulse J = F·Δt equals the resulting change in momentum Δp.
Quick answer
Linear momentum is the product of mass and velocity: p = m·v, measured in kilogram-meters per second (kg·m/s). For example, a 1000 kg car moving at 20 m/s has p = 1000 × 20 = 20,000 kg·m/s. Momentum is a vector, so its direction matches the velocity, and a negative sign indicates motion the opposite way.
Formula & method
p = m · v
- p — Linear momentum (kg·m/s)
- m — Mass of the object (kg)
- v — Velocity of the object (m/s)
Linear momentum equals mass times velocity. In SI units, mass is in kilograms (kg) and velocity in meters per second (m/s), giving momentum in kg·m/s. Momentum is a vector quantity pointing in the same direction as the velocity.
J = F · Δt = Δp
- J — Impulse (N·s, equal to kg·m/s)
- F — Average force applied (N)
- Δt — Time interval the force acts (s)
Impulse equals force times the time interval over which it acts, and by the impulse–momentum theorem it equals the change in momentum (Δp). Force is in newtons (N) and time in seconds (s), so impulse is in N·s, which is dimensionally identical to kg·m/s.
Examples
- Input
- m = 1000 kg, v = 20 m/s
- Result
- p = 20,000 kg·m/s
- Why
- Apply p = m·v directly: p = 1000 kg × 20 m/s = 20,000 kg·m/s. The momentum points in the same direction the car is traveling.
- Input
- m = 0.01 kg (10 g), v = 400 m/s
- Result
- p = 4 kg·m/s
- Why
- p = m·v = 0.01 kg × 400 m/s = 4 kg·m/s. Despite the tiny mass, the very high speed gives the bullet meaningful momentum, which is why it transfers a sharp impulse on impact.
- Input
- F = 50 N, Δt = 4 s
- Result
- J = 200 N·s (Δp = 200 kg·m/s)
- Why
- Use J = F·Δt = 50 N × 4 s = 200 N·s. By the impulse–momentum theorem this equals the change in momentum, so the object's momentum increases by 200 kg·m/s in the direction of the force.
- Input
- m = 75 kg, v = 10 m/s
- Result
- p = 750 kg·m/s
- Why
- p = m·v = 75 kg × 10 m/s = 750 kg·m/s. A 75 kg sprinter at top speed carries more momentum than a 10 g bullet because the much larger mass outweighs the bullet's higher speed.
When to use this tool
- Solving physics problems on linear momentum, collisions, or the impulse–momentum theorem.
- Checking conservation of momentum in collision or recoil scenarios (total p before = total p after).
- Estimating the momentum change delivered by a known force acting for a measured time interval.
- Comparing how mass and velocity each contribute to an object's momentum (e.g. a light fast object vs. a heavy slow one).
Common mistakes
- Confusing momentum with kinetic energy. Momentum is p = m·v (a vector, kg·m/s); kinetic energy is KE = ½mv² (a scalar, joules). Doubling speed doubles momentum but quadruples kinetic energy.
- Forgetting that momentum is a vector. Direction matters — assign a sign convention (e.g. right = +, left = −) before adding the momenta of multiple objects.
- Using non-SI units without converting. Mass must be in kilograms and velocity in m/s. Grams, km/h, or mph must be converted first or the kg·m/s result will be wrong.
- Mixing up impulse and force. Impulse J = F·Δt accounts for how long the force acts; a small force over a long time can deliver the same momentum change as a large force over a short time.
Frequently asked questions
What is the formula for momentum?
Linear momentum is p = m·v, the product of an object's mass (m, in kg) and its velocity (v, in m/s). The result is in kilogram-meters per second (kg·m/s). Because velocity is a vector, momentum is also a vector pointing in the direction of motion.
What are the units of momentum?
The SI unit of momentum is the kilogram-meter per second (kg·m/s). This is equivalent to the newton-second (N·s), which is why impulse (measured in N·s) and momentum share the same dimensions — impulse is a change in momentum.
What is the difference between momentum and kinetic energy?
Momentum (p = m·v) is a vector with units kg·m/s and is linear in velocity. Kinetic energy (KE = ½mv²) is a scalar with units of joules and is proportional to velocity squared. Doubling an object's speed doubles its momentum but quadruples its kinetic energy.
How is impulse related to momentum?
Impulse J equals the average force multiplied by the time it acts: J = F·Δt. The impulse–momentum theorem states that impulse equals the change in momentum: J = Δp = m·Δv. That is why a longer contact time (like a car's crumple zone) reduces the peak force for the same momentum change.
Is momentum always conserved?
Total momentum is conserved in any closed system with no external net force — for example, in collisions or recoil. Momentum can transfer between objects, but the vector sum stays constant. If an outside force (like friction or gravity acting on the system) is present, that force changes the total momentum via impulse.
Can momentum be negative?
Yes. Momentum is a vector, so its sign depends on the chosen direction convention. If you define motion to the right as positive, an object moving left has negative momentum. The sign is essential when adding momenta of objects moving in opposite directions.
Sources & references
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