Impulse required to stop a ball
A 0.50 kg ball moving at 12 m/s stops: J = 0.50(0 − 12) = −6 N·s. The negative sign shows the stopping impulse opposes its motion.
Impulse and Momentum Calculator
Last updated: July 11, 2026
Reviewed by: Starlight Tools science editors
Method: SI-normalized classical mechanics; calculations stay in your browser
Inputs
Use positive and negative values to represent direction along a single axis. Mass and contact time should be positive magnitudes.
Choose a common problem type; the matching fields and solve target will open below.
Momentum mode
Impulse mode
Change in momentum mode
Tip: press Ctrl/Cmd + Enter to calculate.
Results
Ready
Enter values and calculate.
The tool will show solved values, SI-normalized results, and equation notes here.
Momentum uses p = mv. Impulse uses J = FΔt. For a constant-mass one-dimensional model, J = Δp.
Show steps
- Calculate a result to see substitutions, conversions, and arithmetic.
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What this calculator does
Momentum measures how strongly an object keeps moving in a chosen direction. Impulse measures how much a force changes that momentum over time. In simple one-dimensional problems, the two are tied together by the impulse-momentum theorem: the impulse applied to an object equals its change in momentum.
That relationship makes the topic useful across mechanics problems. If you know mass and velocity, you can compute momentum directly. If you know force and contact time, you can compute impulse. If you know initial and final velocity for the same object, you can work out the change in momentum and then infer the average force if the collision or push duration is known. The calculator groups those common textbook pathways into separate modes so you can check arithmetic without switching formulas by hand.
Momentum
p = mv
Use this mode when you know any two of mass, velocity, and momentum.
Impulse
J = FΔt
This mode uses average force over the full contact interval, not an instantaneous force spike.
Change in momentum
Δp = m(vf - vi)
Add contact time if you also want the average force during the change.
Impulse and momentum formulas
| Formula | Use it when | Units |
|---|---|---|
| p = mv | Finding linear momentum from mass and velocity | p: kg·m/s; m: kg; v: m/s |
| m = p/v | Finding mass from momentum and nonzero velocity | kg |
| v = p/m | Finding velocity from momentum and mass | m/s |
| J = FavgΔt | Average force acts over a known contact time | N·s |
| J = Δp = m(vf − vi) | Mass is constant and velocity changes | N·s = kg·m/s |
| Favg = Δp/Δt | Finding average force from momentum change and time | N |
| Δt = J/Favg | Finding contact time from impulse and average force | s |
Variable force: use the integral form J = ∫ F dt; impulse is the signed area under the force–time graph. This calculator’s force-time mode uses a constant average force.
Variables
p is momentum, m mass, v velocity, J impulse, Favg average net force, Δt contact time, and Δp change in momentum. Subscripts i and f mean initial and final.
Advanced: 2D and 3D momentum
Enter mass and velocity components. Leave vz at zero for a 2D vector. Components use px = mvx, and magnitude uses |p| = √(px² + py² + pz²).
p = (6, 8, 0) kg·m/s; |p| = 10 kg·m/s; direction in xy-plane = 53.13°
Common mistakes
Speed is not signed velocity
Choose a positive axis and give motion in the opposite direction a negative velocity.
Use final minus initial
Velocity change is vf − vi. In a rebound, subtracting a negative value increases the magnitude.
Enter mass, not weight
Mass is measured in kg; weight is a force measured in N. Convert weight to mass before using p = mv.
Do not mix raw units
Convert milliseconds, grams, mph, and kN consistently. The calculator does this automatically from each unit menu.
Negative impulse is directional
A negative result usually means momentum changed opposite to your selected positive direction, not that the calculation failed.
Average force is not peak force
J/Δt gives average force across contact. Collision peak force may be much higher.
Assumptions and sign convention
Choose one positive direction before entering numbers. Velocities, momentum, impulse, and average force can be positive or negative depending on that axis. Mass and time intervals should remain positive magnitudes.
This is an educational linear-motion calculator. It does not model rotation, relativistic momentum, variable mass systems, or a force-time curve integral beyond the average-force approximation.
The sign convention matters most in rebound, braking, and thrust problems. A car slowing down while moving in the positive direction has negative impulse because the force changes momentum opposite to the chosen axis. A ball bouncing backward can produce a larger magnitude of impulse than a ball that simply stops, because the final momentum has the opposite sign from the initial momentum. Keeping everything on one axis makes those direction changes visible instead of hiding them inside absolute values.
Worked impulse and momentum examples
Car braking impulse
A 1,500 kg car slows from 20 to 5 m/s: J = 1,500(5 − 20) = −22,500 N·s.
Baseball rebound
A 0.145 kg baseball changes from −35 to 42 m/s: Δp = 0.145(42 − (−35)) = 11.165 kg·m/s.
Average force from contact time
If an impulse of 9.6 N·s acts for 0.08 s, Favg = 9.6/0.08 = 120 N.
Missing velocity from momentum
An 80 kg rider and bicycle with momentum 640 kg·m/s move at v = p/m = 640/80 = 8 m/s.
Momentum and impulse FAQs
What is the difference between momentum and impulse?
Momentum is the motion state of an object at an instant, while impulse is the effect of a force acting over a time interval. The connection is J = Δp.
Why are kg·m/s and N·s equivalent?
A newton is defined as kg·m/s². Multiplying by seconds gives kg·m/s, so the unit for impulse matches the unit for momentum.
Can I enter negative velocities or forces?
Yes. Negative values indicate direction opposite to the positive axis you chose. That is often necessary for rebound and braking problems.
When should contact time stay positive?
Always. Time interval is a duration, so it should be positive. Direction belongs in the force, velocity, impulse, or momentum sign.
Is this enough for engineering sign-off or impact safety analysis?
No. This page is for education and quick estimates. It does not model deformation, stress waves, peak loads, restraint systems, or safety-code requirements.
How do I calculate impulse from force and time?
Multiply the average net force by the contact time: J = FavgΔt. For example, 120 N acting for 0.08 s gives 9.6 N·s.
How do I calculate impulse from mass and velocity change?
For constant mass, use J = m(vf − vi). Preserve velocity signs so the answer includes direction.
How do I find average force from impulse?
Divide impulse by contact time: Favg = J/Δt. This is an interval average, not necessarily the peak force.
Why can impulse be negative?
Impulse is a vector. In a one-dimensional calculation, a negative impulse points opposite to the positive axis you selected.
Is impulse a vector?
Yes. Impulse has the same direction as the change in momentum and the net force that causes it.
What units does impulse use?
The SI unit is the newton-second (N·s), exactly equivalent to kg·m/s. US customary problems may use lbf·s.
What does contact time mean?
Contact time is the duration over which the force acts, such as the brief interval when a bat touches a ball or a tire brakes against the road.
When should I use Δp instead of FΔt?
Use Δp when mass and initial/final velocities are known. Use FΔt when average force and duration are known. The impulse-momentum theorem says both routes produce the same impulse for the same event.
Methodology and references
The calculator converts each entry to SI units, applies classical linear momentum and the impulse-momentum theorem, then converts the requested output back to its selected unit. It assumes constant mass and uses average net force over the entered interval. Displayed values are rounded to six decimal places, while calculations retain JavaScript floating-point precision.
Definitions follow standard introductory mechanics and SI unit relationships. References: OpenStax Physics: Linear Momentum, Force, and Impulse and the BIPM SI Brochure.