Impulse and Momentum Calculator finds impulse from mass and velocities, force and time, or two momentum values. It also shows the average force, momentum and kinetic energy change.
Pick the direction you call positive. A velocity the other way gets a minus sign.
More Inputs and Units
Step-by-Step Working
- Known values in SI units: m = 0.145 kg, v₁ = −40 m/s, v₂ = 45 m/s, Δt = 0.002 s
- Initial momentum: p₁ = m × v₁ = 0.145 × (−40) = −5.8 kg·m/s
- Final momentum: p₂ = m × v₂ = 0.145 × (45) = 6.53 kg·m/s
- Impulse: J = p₂ − p₁ = 6.53 − (−5.8) = 12.33 N·s
- Average force: F = J ÷ Δt = 12.33 ÷ 0.002 = 6,162.5 N
- Result: J = 12.33 N·s, delivered by an average 6,162.5 N over 2 ms
Force at Other Contact Times
| Contact Time | Average Force | Compared |
|---|---|---|
| 0.5 ms | 24,650 N | 4 times the force |
| 1 ms | 12,325 N | 2 times the force |
| 2 ms | 6,162.5 N | Your result |
| 4 ms | 3,081.3 N | 1/2 of the force |
| 8 ms | 1,540.6 N | 1/4 of the force |
| 20 ms | 616.25 N | 1/10 of the force |
Force-Time Graph
Impulse and Momentum Calculator Settings
The Impulse and Momentum Calculator starts from whichever pair of values a problem gives. The Known Values menu has seven settings. Mass and Velocities is the usual one, and the others take a force and a time, several force stages, two momentum values, an impulse with a time or a force, or two velocities at angles.
Direction matters in every setting. Pick one direction as positive, and give anything pointing the other way a minus sign. The default is a 0.145 kg baseball arriving at −40 m/s and leaving at 45 m/s, which is an impulse of 12.33 N·s.
Force and Time opens with a 50 N push lasting 0.2 s on a 2 kg object already moving at 1.5 m/s. That is 10 N·s of impulse, a velocity change of 5 m/s and a final momentum of 13 kg·m/s. The mass and starting velocity sit under More Inputs.
Four cards sit under the headline. They show the momentum before and after, the average force for the contact time entered, the kinetic energy gained or lost, and the same results in US units. A table then shows the force the same impulse would need over shorter and longer contact times.
Impulse and Momentum Formulas
Momentum is mass times velocity, written p. Impulse is the average force on an object times the time that force acts, written J. The OpenStax College Physics text defines both and links them through Newton’s second law.
$$p = m v$$
$$J = F \Delta t$$
The impulse-momentum theorem says the impulse on an object equals its change in momentum. This is the impulse and momentum definition most courses test, and it gives a second formula for impulse that needs no force at all.
$$J = \Delta p = m (v_2 – v_1)$$
Setting the two formulas equal turns a momentum change into a force. That is the route from momentum to force, and it is how the Impulse and Momentum Calculator fills in the average force card for any contact time.
$$F = \frac{\Delta p}{\Delta t}$$
So there are three possible formulas for impulse, and they always agree. Force times time, final momentum minus initial momentum, and mass times velocity change all give the same number for the same event.
Impulse With Mass and Velocity
To calculate impulse with mass and velocity, multiply the mass by the change in velocity. For the default baseball that is 0.145 × (45 − (−40)) = 12.33 N·s. The velocity change is 85 m/s, not 5 m/s, because the ball reverses direction.
That reversal is the most common slip in these problems. Turning an object around takes more impulse than stopping it, since the momentum change covers both the stop and the return. The message under the results points this out whenever the two velocities have opposite signs.
A textbook case checks the numbers. The OpenStax example of a tennis serve uses a 0.057 kg ball leaving the racquet at 58 m/s after 5.0 ms of contact. Its answers are 3.306 kg·m/s and 661 N, and the Impulse and Momentum Calculator returns 3.31 N·s and 661.2 N.
The same setting works as a change in momentum calculator. The first card lists the initial momentum, the final momentum and the velocity change, so the impulse can be read as the gap between the two momentum values.
Momentum to Force and Contact Time
An impulse fixes the product of force and time, not either one alone. A 70 kg person moving at 15 m/s has 1,050 kg·m/s of momentum, and stopping takes 1,050 N·s however it happens. The time it takes decides the force.
| Stopping time | Average force | Deceleration |
|---|---|---|
| 5 ms | 210,000 N | 305.9 g |
| 50 ms | 21,000 N | 30.59 g |
| 100 ms | 10,500 N | 15.3 g |
| 500 ms | 2,100 N | 3.06 g |
The contact time table in the Impulse and Momentum Calculator shows the same pattern for any entry. Ten times the stopping time means one tenth of the force. Airbags, crumple zones, padded floors and bent knees on landing all work this way. They cannot change the impulse, so they stretch the time.
The Impulse and Force setting runs this backward. With an impulse of 1,050 N·s and a force limit of 5,000 N, the contact has to last at least 0.21 s. The speed at the start of such a stop can come from the Free Fall Calculator, which gives the impact velocity of a drop.
Impulse and Momentum Calculator With Angle
When the object leaves in a new direction, momentum has to be subtracted as a vector. The Velocities at Angles setting takes a speed and a direction before and after, with angles measured counterclockwise from the positive x direction.
Take the same baseball arriving at 40 m/s along 180° and leaving at 45 m/s along 30°. The impulse is 11.45 N·s along x and 3.26 N·s along y. Together that is 11.91 N·s pointing at 15.9°, and an average force of 5,953.3 N over 2 ms.
This is slightly less than the straight-back hit, because the ball does not fully reverse. Problems with two objects that collide and share momentum belong to the Conservation of Momentum Calculator, which solves for the velocities after the collision.
Momentum Examples
Ten examples of momentum show how mass and speed trade off. The masses and speeds are round figures chosen for illustration, and each momentum is simply their product. Any row can be entered in the Impulse and Momentum Calculator as a mass and a final velocity.
| Moving object | Mass | Speed | Momentum |
|---|---|---|---|
| Tennis ball on a serve | 0.057 kg | 58 m/s | 3.31 kg·m/s |
| Rifle bullet | 0.01 kg | 400 m/s | 4 kg·m/s |
| Pitched baseball | 0.145 kg | 40 m/s | 5.8 kg·m/s |
| Bowling ball | 7 kg | 8 m/s | 56 kg·m/s |
| Adult walking | 70 kg | 1.4 m/s | 98 kg·m/s |
| Sprinter | 70 kg | 10 m/s | 700 kg·m/s |
| Cyclist with bike | 85 kg | 10 m/s | 850 kg·m/s |
| Small car | 1,200 kg | 25 m/s | 30,000 kg·m/s |
| Loaded truck | 10,000 kg | 20 m/s | 200,000 kg·m/s |
| Freight train | 5,000,000 kg | 10 m/s | 50,000,000 kg·m/s |
A bullet and a baseball carry similar momentum even though one is far lighter, because the bullet is ten times faster. A walking adult outweighs both in momentum. The train at the bottom is slow, yet nothing on the list is harder to stop.
Impulse and Momentum Questions
Is impulse the same as momentum?
They are related but not the same. Momentum is something a moving object has at one instant. Impulse is what a force delivers over a stretch of time, and it equals the change in momentum. Momentum is not also known as impulse, though the two share a unit.
A ball flying at steady speed has momentum and receives no impulse. A ball hit by a bat receives an impulse, and its momentum changes by exactly that amount. Converting momentum to impulse just means subtracting the initial momentum from the final one.
What is the SI unit of momentum and impulse?
The SI unit of momentum is the kilogram-meter per second, kg·m/s. The SI unit of impulse is the newton-second, N·s. They are the same unit written two ways, since one newton is one kg·m/s², so 12.33 N·s and 12.33 kg·m/s are equal.
Impulse is not a joule. A joule is a unit of energy, equal to a newton times a meter, while impulse is a newton times a second. The third card of the Impulse and Momentum Calculator keeps the two apart by showing kinetic energy in joules beside the impulse in newton-seconds.
What is momentum in simple terms?
Momentum is how hard something is to stop. A heavier object or a faster one has more of it. For a child, a rolling bowling ball and a rolling tennis ball at the same speed make the point, since only one of them knocks the pins over.
Physics books also call it linear momentum, and older texts call it quantity of motion. Its short form is the letter p. The rule for momentum is that it stays constant unless an outside force acts, and that force changes it by the impulse it delivers.
How do you calculate total impulse?
Add the impulse of each stage, keeping its sign. The Several Force Stages setting does this for up to three stages. Its default is a 50 N push for 0.2 s followed by a 20 N force the other way for 0.1 s, which is 10 − 2 = 8 N·s of total impulse.
The same result could come from one steady force of 26.67 N over the full 0.3 s, and the second card shows that average. For the 2 kg object the velocity rises by 4 m/s and the final momentum is 11 kg·m/s. On the force-time graph, the total impulse is the bar areas added together, with bars below the line counted as negative.
A resisting force always enters as a negative stage in the Impulse and Momentum Calculator. A sliding box slowed by friction is the common case, and the size of that force comes from the Friction Calculator.