Car Collision Physic

Question

A 40,000 kg car collides with a stationary car. They couple together, and 35 percent of the initial kinetic energy is dissipated as heat, sound, vibrations, and so on. What is the mass of the car?

Answer 1

This is incorrect - this is an inelastic collision, so the initial kinetic energy is *not* equal to the final kinetic energy and the first equation does not hold.

What does hold is conservation of momentum, in all collisions, regardless of if they are elastic, inelastic, or anything in between:

m1*v1 = (m1 + m2)*vf (equation 1)

where v1 is the first car's initial velocity and vf is the final velocity of both cars.

There are 2 unknowns: the mass of the stationary car (m2) and the final velocity vf. So, we need a 2nd equation. That comes from the energy information that we are given: 35% of the initial kinetic energy is dissipated. In equation form that is written as:

0.65 * KEi = KEf

0.65 * 1/2 * m1 * v1^2 = 1/2 * (m1 + m2) * vf^2 (equation 2)

(Note that in the previous answer the 0.65 is on the wrong side of the equation.)

Rearranging and simplifying, we get:

m2 = m1(1/0.65-1) = 21,538 ~ 21,500 (rounded to 3 sig figs)

Answer 2

We can use the law of conservation of energy to solve this problem. Since the two cars couple together and move as one after the collision, the initial kinetic energy is equal to the final kinetic energy. We can write:

1/2 * m1 * v1^2 = 1/2 * (m1 + m2) * v^2

where m1 is the mass of the first car, v1 is its initial velocity, m2 is the mass of the second car (which is initially stationary), and v is their common velocity after the collision.

We are given that m1 = 40,000 kg and that 35 percent of the initial kinetic energy is dissipated. This means that the final kinetic energy is only 65 percent of the initial kinetic energy. We can write:

1/2 * m1 * v1^2 = 0.65 * 1/2 * (m1 + m2) * v^2

Simplifying and rearranging, we get:

m2 = m1 * (1 - 0.65) / 0.65

m2 = 40,000 * 0.35 / 0.65

m2 = 21,621.54 kg

Therefore, the mass of the stationary car is approximately 21,622 kg.