Impulse and momentum are physical concepts that are easily seen from Newton’s Laws of Motion.

Start with this equation of motion for constant acceleration.

v = v_{0} + at

where

v = velocity

v_{0} = initial velocity

a = acceleration

t = time

If you rearrange the equation, you get

v – v_{0} = at

Newton’s second law deals with force.

F = ma

where

F = force

m = mass

a = acceleration

solve this for a and get

a = F/m

Stick this into the velocity equation and get

v – v_{0} = (F/m)t

Multiply both sides by m

mv – mv_{0} = Ft

The left side of the equation deals with momentum (often denoted by a lower-case p) and the right side is impulse (often denoted by an upper-case letter J).

Mass times velocity is known as momentum and force applied over time is called impulse.

### Impulse and Momentum Example Problem

**Question:** A 50 kg mass is sitting on a frictionless surface. An unknown constant force pushes the mass for 2 seconds until the mass reaches a velocity of 3 m/s.

a) What is the initial momentum of the mass?

b) What is the final momentum of the mass?

c) What was the force acting on the mass?

d) What was the impulse acting on the mass?

**Part a)** What is the initial momentum?

Momentum is mass times velocity. Since the mass is at rest, the initial velocity is 0 m/s.

momentum = m⋅v = (50 kg)⋅(0 m/s) = 0 kg⋅m/s

**Part b)** What is the final momentum?

After the force is finished acting on the mass, the velocity is 3 m/s.

momentum = m⋅v = (50 kg)⋅(3 m/s) = 150 kg⋅m/s

**Part c)** What was the force acting on the mass?

mv – mv_{0} = Ft

From parts a and b, we know mv_{0} = 0 kg⋅m/s and mv = 150 kg⋅m/s.

150 kg⋅m/s – 0 kg⋅m/s = Ft

150 kg⋅m/s = Ft

Since the force was in effect over 2 seconds, t = 2 s.

150 kg⋅m/s = F ⋅ 2s

F = (15 kg⋅m/s) / 2 s

F = 75 kg⋅m/s^{2}

**Unit Fact:** **kg⋅m/s ^{2}** can be denoted by the derived SI unit

**Newton**(symbol

**N**)

F = 75 N

**Part d)** What was the impulse acting on the mass?

The impulse is the force multiplied by the time passed. It is also equal to the change in momentum over the same time period.

Ft = 75 N ⋅ 2 s

Ft = 150 Ns or 150 kg⋅m/s

The impulse was 150 kg⋅m/s.

These problems are relatively simple as long as you keep your units straight. Impulse and momentum should have the same units: mass⋅velocity or force⋅time. Check your units when you check your answer.

Another possible way to cause errors is to confuse your vector directions. Velocity and Force are both vector quantities. In this example, the mass was pushed in the direction of the final velocity. If another force pushed in the opposite direction to slow down the mass, the force would have a negative value compared to the velocity vector.

If you found this helpful, check out other Physics Example Problems.

t = 2s as defined in the problem. Parts (c) & (d) have t=3s. so (c) should be 75 N while (d) is still 150 Ns

Thanks for pointing out the error. I’ve corrected it with the correct time value.

Time is 2 seconds and velocity is 3m/s. Not the other way around