The impulse calculator above works the impulse-momentum theorem in any direction. Impulse is the product of a force and the time it acts, and it equals the change in momentum it produces: J = FΔt = mΔv. That single equation contains four quantities, and real problems come in every combination of three known and one unknown. This tool solves for the average force, the contact time, the mass, or the velocity the object ends up with.
Arb Digital publishes free physics calculators that respect signs rather than quietly taking absolute values, and impulse is the place where that matters most. A ball caught and held changes momentum by one amount; the same ball struck back the way it came changes momentum by more than twice as much. Any tool that treats velocities as magnitudes gets the second case badly wrong, so every velocity field here is signed and the result reports the direction as well as the size.
What This Impulse Calculator Does
You choose which of the four terms is unknown, and the panel adjusts so you only enter the three you have. Solving for force is the classic collision question: something changed speed in a known time, what force did that take. Solving for time inverts it: you know what force a structure or a body can tolerate and want the contact duration that keeps you under it. Solving for mass is how a measured impulse identifies an unknown projectile, and solving for final velocity answers what a thruster, a kick or a bat does to an object over a known burn or contact.
The grid always shows four things regardless of mode. The impulse itself, in newton-seconds, is the quantity the whole theorem is about. The momentum before and after are shown separately so you can see the sign change directly when a rebound occurs. The average acceleration is reported in both metres per second squared and multiples of g, because that is the figure that matters when a person or an instrument is inside the object.
Degenerate cases are handled explicitly rather than being allowed to produce nonsense. A zero velocity change with a non-zero force has no consistent solution; a zero force cannot produce a velocity change in any finite time; a zero contact time implies an infinite force. Each of those gets a message explaining the physics rather than a NaN.
How to Use It
- Pick the unknown first. The input panel hides the field you are solving for, so you can never accidentally over-specify the problem.
- Choose a positive direction and keep it. If the ball arrives moving left, enter its initial velocity as negative and its rebound as positive, or the other way round consistently. The two must not both be positive unless the object genuinely never changed direction.
- Use the average force, not the peak. The theorem is exact for the average over the contact. Real force curves rise and fall, and the peak is typically well above the mean.
- Enter contact time in seconds. Most impacts are milliseconds, so a racket contact is 0.005 rather than 5. This is the single most common data-entry error on this page.
- Compare the before and after momentum figures. If they have opposite signs the object reversed, and the impulse is the full distance between them rather than the difference in magnitudes.
The Formula: How Impulse Is Calculated
The impulse-momentum theorem says that the impulse applied to an object equals the change in that object's momentum. In symbols, J = Favg × Δt = mvf − mvi. OpenStax University Physics Volume 1, section 9.2 on impulse and collisions, derives it directly from Newton's second law and works through why extending the collision time reduces the force experienced. Momentum itself, p = mv, is defined in section 9.1 on linear momentum of the same text.
Work the defaults through, which describe a tennis ball being returned. The ball has a mass of 0.058 kg, arrives at 25 m/s and leaves at 30 m/s in the opposite direction, so with the outgoing direction taken as positive the initial velocity is −25 m/s and the final is +30 m/s. The velocity change is 30 − (−25) = 55 m/s, not 5 m/s. The impulse is 0.058 × 55 = 3.19 N·s. Over a contact time of 0.005 s the average force is 3.19 ÷ 0.005 = 638 N, roughly the weight of a 65 kg person applied for five thousandths of a second.
The momentum figures make the sign issue concrete. Before the hit the momentum is 0.058 × (−25) = −1.45 kg·m/s; after it is 0.058 × 30 = +1.74 kg·m/s. The impulse spans the whole interval between them. Had the racket merely stopped the ball dead, the impulse would have been only 1.45 N·s and the force only 290 N. Reversing the ball costs more than twice as much as absorbing it, which is why blocking a shot is so much easier than driving one back.
Why Bouncing Hurts More Than Sticking
This is the most useful and least intuitive consequence of the theorem, and it applies far beyond sport. Consider an object arriving with momentum p. If it stops dead, the momentum change is p. If it rebounds elastically at the same speed, the momentum change is 2p, so the impulse and therefore the average force are doubled for the same contact time.
That is why a hailstone bouncing off a roof panel loads it harder than wet snow of the same mass landing and staying. It is why a rubber-headed mallet delivers more impulse to a workpiece than a dead-blow hammer filled with shot, which is designed precisely to avoid rebound. It is also why the sail of a solar sail made of reflective film gets twice the thrust of a black absorbing one: reflected photons reverse their momentum instead of merely surrendering it.
The tool shows this directly through the momentum before and after figures. Whenever those two have opposite signs, the impulse is larger than either one alone. It is worth running the bounce preset and then setting the final velocity to zero to watch the required force roughly halve while nothing else changes.
Extending the Time Is the Whole of Safety Engineering
For a fixed momentum change, force and contact time are inversely proportional. There is no way to reduce the impulse an object must absorb when it stops — that is set by its mass and speed — but there is a great deal of scope to change how long the stopping takes. Every result of that trade is familiar. Bending your knees on landing, rolling with a fall, boxing gloves, crash helmets, air bags, crumple zones and packaging foam all stretch the same fixed impulse over a longer interval, and the force falls in exact proportion.
The arithmetic is unforgiving in the other direction too. Halving the contact time doubles the force. A car striking a rigid concrete wall stops in a fraction of the time it would take stopping against a deformable barrier, and the occupants experience the difference as a directly proportional increase in deceleration. Our impact force calculator runs the same collision from the distance side rather than the time side, which is often the easier quantity to estimate for a crumple or a foam pad.
Impulse and Energy Are Not the Same Thing
Both describe a collision, and confusing them causes real errors. Impulse is force times time and equals the momentum change; kinetic energy is force times distance and equals the work done. Momentum is conserved in every collision. Kinetic energy is only conserved in a perfectly elastic one, and in most real impacts a large fraction of it becomes heat and permanent deformation.
The practical difference shows up in what each one predicts. Momentum tells you what velocities result from a collision, which is why the collision momentum calculator uses it to solve two-body impacts. Energy tells you how much damage the collision can do, which is why the kinetic energy calculator is the right tool for that question. A heavy slow object and a light fast one can have identical momentum and very different energies, and the two would behave quite differently on arrival.
Where This Sits Next to Our Momentum Calculator
The boundary is worth stating precisely, because the two pages use the same physics for different jobs. Our momentum calculator is built around p = mv: it solves for momentum, mass or velocity, and it will show you the force needed to stop an object in a time you specify. It always works forwards, from a moving object to rest.
This page inverts that relationship and generalises it. It solves for contact time given a force limit, for mass given a measured impulse, and for a final velocity that need not be zero and may be in the opposite direction. If your object ends up moving — bounced, deflected, or driven back the way it came — this is the page that handles it correctly. If you simply want p = mv, use the momentum calculator. For the acceleration behind the force, the acceleration calculator and the net force calculator cover Newton's second law in its instantaneous form.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering both velocities as positive after a rebound — a reversal means opposite signs, and getting this wrong understates the impulse by more than a factor of two.
- Typing contact time in milliseconds — the field expects seconds, so a five-millisecond contact is 0.005. Entering 5 makes the force a thousand times too small.
- Treating the answer as a peak force — the theorem gives the average over the contact, and real peaks run considerably higher.
- Using impulse to predict damage — damage tracks energy, not momentum, and the two rank collisions differently.
- Forgetting other forces during a long contact — over a contact of a second or more, gravity and drag contribute their own impulse, and this single-force model no longer describes the whole change.
Related Free Tools From Arb Digital
For plain p = mv, use the momentum calculator, and for a two-body collision where both objects move afterwards use the collision momentum calculator. The distance-based view of the same impact is on the impact force calculator. For the energy side, see the kinetic energy calculator, and for the Newton's-second-law view see the net force calculator and the acceleration calculator. The force converter translates newtons into pounds-force and kilogram-force when a specification uses different units. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
Impulse is a force multiplied by the time it acts, measured in newton-seconds. It equals the change in an object's momentum, which is why a newton-second and a kilogram-metre per second are the same unit written two ways.
Because momentum is a vector. If an object reverses direction, its initial and final velocities have opposite signs and the velocity change is their sum rather than their difference. Treating both as positive magnitudes understates the impulse by more than half in a typical rebound.
Stopping an object removes its momentum. Bouncing it removes that momentum and then supplies an equal amount in the opposite direction, so the total change is roughly doubled. Over the same contact time, that doubles the average force.
No, it is the average over the contact. In a real collision the force starts at zero, rises to a peak and falls back, and the peak is commonly one and a half to three times the average. Use the average for momentum bookkeeping and measured data for peak checks.
They lengthen the contact time. The impulse required to stop an occupant is fixed by their mass and speed, but spreading it over a longer interval reduces the average force in direct proportion, and the deceleration in g falls with it.
No. Impulse is force times time and tracks momentum, which is conserved in every collision. Kinetic energy is force times distance and is conserved only in perfectly elastic collisions. Two objects can carry equal momentum and very different energies.
The momentum calculator solves p = mv and works forwards from a moving object to rest. This page solves the impulse-momentum theorem in any direction, including for contact time or mass, and it handles a final velocity that is non-zero or reversed.
This tool is provided for educational and study use. It models a single average force acting over a stated contact time and makes no allowance for other forces during contact, for peak force, or for any injury or safety threshold.
