Formula for Change in Momentum: Final Momentum Minus Initial Momentum, Equal to Mass Times Change in Velocity and to Impulse
Keyword: formula change in momentum | Updated October 16, 2026
For the related system-level rule in collisions, see Conservation of Momentum: Formula, 3 Collision Types, and 5 Solved Examples.
The formula for change in momentum is Δp = pf – pi = m(vf – vi) = mΔv, for an object with constant mass. Change in momentum equals impulse, J = FavgΔt. The SI unit is kg·m/s. This article covers 5 equivalent formulas, sign rules, 7 worked examples, applications, and common mistakes.
What Is the Formula for Change in Momentum?
The formula for change in momentum is Δp = pf – pi = m(vf – vi) = mΔv. According to OpenStax University Physics, the product mΔv is the change in momentum Δp. Momentum is the product of mass and velocity, p = mv. Momentum is a vector with the same direction as the velocity.
The table below defines each symbol in the change in momentum formulas.
| Symbol | Quantity | SI Unit | Definition |
|---|---|---|---|
| Δp | Change in momentum | kg·m/s | Final momentum minus initial momentum |
| p, pi, pf | Momentum, initial momentum, final momentum | kg·m/s | Mass times velocity |
| m | Mass | kg | Amount of matter in the object |
| vi, vf | Initial velocity, final velocity | m/s | Velocity before and after the interaction |
| Δv | Change in velocity | m/s | Final velocity minus initial velocity |
| J | Impulse | N·s | Force multiplied by the time it acts |
| Favg | Average net force | N | Net force averaged over the time interval |
| Δt | Time interval | s | Duration of the interaction |
What Are the 5 Equivalent Formulas for Change in Momentum?
Change in momentum has 5 equivalent formulas. The table below lists each formula with its condition of use.
| No. | Name | Formula | Condition of Use |
|---|---|---|---|
| 1 | Definition | Δp = pf – pi | Initial and final momentum known |
| 2 | Constant mass | Δp = m(vf – vi) = mΔv | Mass stays constant and velocities known |
| 3 | Impulse-momentum theorem | Δp = J = FavgΔt | Average force and time known |
| 4 | Integral form | Δp = ∫F dt | Force changes with time |
| 5 | Component form | Δpx = m(vfx – vix), Δpy = m(vfy – viy) | Motion in two dimensions |
What Is the Unit of Change in Momentum?
The SI unit of change in momentum is kilogram meter per second (kg·m/s). The newton second (N·s) is an equivalent unit. One newton second equals 1 kg·m/s² multiplied by 1 s, which gives 1 kg·m/s.
How Does the Sign Convention Work for Change in Momentum?
Change in momentum uses a chosen positive direction. Velocity toward the positive direction is positive, and velocity toward the negative direction is negative. The sign of Δp shows the direction of the net impulse.
A 0.5 kg ball hits a wall at +10 m/s and rebounds at -8 m/s. The change in momentum is 0.5 × (-8 – 10) = -9 kg·m/s. The negative sign shows that the wall pushes the ball in the negative direction.
How Is Change in Momentum Related to Force and Time?
Change in momentum equals impulse, J = Δp = FavgΔt. OpenStax University Physics defines impulse as the product of a force and the time interval over which the force acts. The impulse-momentum theorem states that the impulse equals the change in momentum.
The average force equals the change in momentum divided by the time interval, Favg = Δp / Δt. Newton’s second law in momentum form states that the net force equals the rate of change of momentum, Fnet = dp/dt.
The list below summarizes 3 graph relationships.
- Area under a force-time graph represents impulse.
- Impulse represents change in momentum.
- Height of a rectangle with the same area and base represents average force.
How Do You Calculate Change in Momentum Step by Step?
To calculate change in momentum, subtract initial momentum from final momentum. Follow these 6 steps:
- Choose a positive direction.
- Write the initial and final velocities with signs.
- Convert the mass to kilograms.
- Subtract the initial velocity from the final velocity to get Δv.
- Multiply the mass by Δv.
- State the result with a unit and a direction.
What Are Examples of Change in Momentum Calculations?
The table below shows 7 worked examples. Each row lists the scenario, known values, calculation, and result.
| No. | Scenario | Known Values | Calculation | Result |
|---|---|---|---|---|
| 1 | Car speeds up | m = 1,000 kg, vi = 10 m/s, vf = 25 m/s | 1,000 × (25 – 10) | +15,000 kg·m/s |
| 2 | Ball rebounds from a wall | m = 0.5 kg, vi = +10 m/s, vf = -8 m/s | 0.5 × (-8 – 10) | -9 kg·m/s |
| 3 | Force stops a sliding block | Favg = 15 N, Δt = 0.24 s | 15 × 0.24 | 3.6 kg·m/s |
| 4 | Force from change in momentum | Δp = 12 kg·m/s, Δt = 0.04 s | 12 / 0.04 | 300 N |
| 5 | Bat hits a baseball | m = 0.15 kg, vi = -40 m/s, vf = +50 m/s | 0.15 × (50 – (-40)) | +13.5 kg·m/s |
| 6 | Two carts stick together | Cart A: 2 kg at +3 m/s. Cart B: 1 kg at rest. Final speed: 2 m/s | A: 2 × (2 – 3). B: 1 × (2 – 0) | A: -2 kg·m/s. B: +2 kg·m/s. Total: 0 |
| 7 | Ball changes direction in two dimensions | m = 0.2 kg, vi = (4, 0) m/s, vf = (0, 3) m/s | 0.2 × (-4, 3) | (-0.8, 0.6) kg·m/s, magnitude 1.0 kg·m/s at 143° from the +x axis |
Example 3 follows the block problem in OpenStax College Physics, where a 15 N force acting for 0.24 s gives a change in momentum of 3.6 kg·m/s. Example 6 shows that the changes in momentum of the 2 carts are equal and opposite.
Why Does a Longer Collision Time Reduce the Force?
A longer collision time reduces the average force because the change in momentum stays the same, Favg = Δp / Δt. OpenStax College Physics states that an occupant has the same change in momentum with or without an air bag, but the force is much smaller when it acts over a longer time.
The table below shows an example with a 60 kg occupant stopping from 15 m/s, which gives Δp = -900 kg·m/s.
| Stopping Method | Stopping Time | Average Force |
|---|---|---|
| Rigid dashboard | 0.02 s | 45,000 N |
| Air bag | 0.10 s | 9,000 N |
What Is the Difference Between Momentum, Change in Momentum, and Impulse?
Momentum describes motion at one moment, change in momentum describes the difference between 2 moments, and impulse describes the cause of that difference. The table below compares the 3 quantities with kinetic energy.
| Quantity | Symbol | Formula | SI Unit | Type |
|---|---|---|---|---|
| Momentum | p | p = mv | kg·m/s | Vector |
| Change in momentum | Δp | Δp = m(vf – vi) | kg·m/s | Vector |
| Impulse | J | J = FavgΔt | N·s | Vector |
| Kinetic energy | K | K = ½mv² | J | Scalar |
How Does Change in Momentum Relate to Conservation of Momentum?
In an isolated system, the total change in momentum is zero, so Δp₁ = -Δp₂ for 2 interacting objects. Internal forces transfer momentum between objects without changing the total. External forces change the total momentum of the system, according to the impulse-momentum theorem.
Where Is Change in Momentum Used in Real Life?
Change in momentum applies to 5 main fields.
- Automotive safety: designing air bags, seat belts, and crumple zones to lengthen collision time
- Sports science: analyzing bat, racquet, and club contact with a ball
- Aerospace engineering: calculating rocket thrust from the momentum of exhaust gas
- Protective equipment: designing helmets that extend impact time
- Accident reconstruction: estimating collision forces from changes in velocity
What Mistakes Occur When Calculating Change in Momentum?
Five mistakes cause most errors.
- Using speed instead of velocity
- Ignoring the sign of velocity after a rebound
- Subtracting in the wrong order, initial minus final
- Mixing grams with kilograms
- Confusing impulse with force
What Do People Also Ask About the Formula for Change in Momentum?
What is the formula for change in momentum?
The formula for change in momentum is Δp = m(vf – vi), if mass is constant.
What is the unit of change in momentum?
The unit of change in momentum is kg·m/s, which equals N·s.
What is the relationship between impulse and change in momentum?
Impulse equals change in momentum, J = Δp.
How do you find force from change in momentum?
The average force equals the change in momentum divided by the time interval, Favg = Δp / Δt.
Can change in momentum be negative?
Yes, change in momentum is negative when the net impulse points in the negative direction.
What is the change in momentum when an object stops?
The change in momentum is Δp = -mvi, because the final momentum is zero.
What is the change in momentum when a ball rebounds?
The change in momentum is Δp = -m(vi + |vf|), if the incoming direction is positive.
Is change in momentum a vector?
Yes, change in momentum is a vector with magnitude and direction.
What is the change in momentum of an isolated system?
The total change in momentum of an isolated system is zero.
How does collision time affect force?
A longer collision time reduces the average force, if the change in momentum stays the same.
What Is the Formula for Change in Momentum in Summary?
The formula for change in momentum is Δp = pf – pi = m(vf – vi) = mΔv. The impulse-momentum theorem gives the equivalent form Δp = J = FavgΔt. The SI unit is kg·m/s, which equals N·s. Change in momentum is a vector, so the sign depends on the chosen positive direction. A longer collision time lowers the average force for the same change in momentum. The total change in momentum of an isolated system is zero.
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