A collision is considered an inelastic collision when kinetic energy is lost during the collision. This inelastic collision example problem will show how to find the final velocity of a system and the amount of energy lost from the collision.

### Inelastic Collision Example Problem

**Question:** A 3000 kg truck travelling at 50 km/hr strikes a stationary 1000 kg car, locking the two vehicles together.

A) What is the final velocity of the two vehicles?

B) How much of the initial kinetic energy is lost to the collision?

Solution:

Part A: To find the final velocity, remember momentum is conserved before and after the collision.

total momentum before = total momentum after

m_{T}v_{T} + m_{C}v_{C} = (m_{T} + m_{C})v_{Final}

where

m_{T} = mass of the truck = 3000 kg

m_{C} = mass of the car = 1000 kg

v_{T} = velocity of the truck = 50 km/hr

v_{C} = velocity of the car = 0 km/hr

v_{Final} = final velocity of the combined truck and car = ?

Plug these values into the equation

(3000 kg)(50 km/hr) + (1000 kg)(0 km/hr) = (3000 kg + 1000 kg)v_{Final}

Solve for v_{Final}

150,000 kg⋅km/hr + 0 kg⋅km/hr = (4000 kg)v_{Final}

150,000 kg⋅km/hr = (4000 kg)v_{Final}

v_{Final} = 150,000 kg⋅km/hr/(4000 kg)

v_{Final} = 37.5 km/hr

The final velocity of the combined truck-car mass continues on at 37.5 km/hr.

Part B: To find the amount of kinetic energy lost in the collision, we need to find the kinetic energy just before the collision and after the collision.

Kinetic energy before = ½m_{T}v_{T}^{2} + ½m_{C}v_{C}^{2}

KE before = ½(3000 kg)(50 km/hr)^{2} + ½(1000 kg)(0 km/hr)^{2}

KE before = ½(3000 kg)(50 km/hr)^{2}

Let’s leave it at that for right now. Next, we need to find the final kinetic energy.

Kinetic energy after = ½(m_{T} + m_{C})v_{Final}^{2}

KE after = ½(4000 kg)(37.5 km/hr)^{2}

Divide KE after by KE before to find the ratio between the values.

Working this out, we get

^{KEafter}/_{KE before} = ^{3}/_{4}

^{3}/_{4} of the total kinetic energy of the system remains after the collision. This means ^{1}/_{4} of the energy is lost to the collision.