Question

Two objects moving with a speed v travel in opposite directions in a straight line. The objects stick together when they collide, and move with a speed of v/2 after the collision.

Required:
a. What is the ratio of the final kinetic energy of the system to the initial kinetic energy?
b. What is the ratio of the mass of the more massive object to the mass of the less massive object?

Answer 1

(a) The ratio is 1 : 4.

(b) The ratio is 1 : 3.

Step-by-step explanation:

Let the mass of each object is m and m'.

They initially move with velocity v opposite to each other.

Use conservation of momentum

m v - m' v = (m + m') v/2

2 (m - m') = (m + m')

2 m - 2 m' = m + m'

m = 3 m' .... (1)

(a) Let the initial kinetic energy is K and the final kinetic energy is K'.

`K = 0.5 mv^2 + 0.5 m' v^2 \\\\K = 0.5 (m + m') v^2..... (1)`

`K' = 0.5 (m + m') (v^2)/(4).... (2)`

The ratio is

K' : K = 1 : 4

(b) m = 3 m'

So, m : m' = 3 : 1

Answer 2

Step-by-step explanation:

Let the mass of objects be m₁ and m₂ .

Total kinetic energy = 1/2 m₁ v² + 1/2 m₂ v²= 1/2 ( m₁ + m₂ ) v²

Total kinetic energy after collision= 1/2 ( m₁ + m₂ ) v² / 4 = 1/2 ( m₁ + m₂ ) v² x .25

final KE / initial KE = 1/2 ( m₁ + m₂ ) v² x .25 / 1/2 ( m₁ + m₂ ) v²

= 0.25

b )

Applying law of conservation of momentum to the system . Let m₁ > m₂

m₁ v - m₂ v = ( m₁ + m₂ ) v / 2

m₁ v - m₂ v = ( m₁ + m₂ ) v / 2

m₁ - m₂ = ( m₁ + m₂ ) / 2

2m₁ - 2 m₂ = m₁ + m₂

m₁ = 3m₂

m₁ / m₂ = 3 / 1