The first car has twice the mass of a second car, but only half as much kinetic energy.

Question

The first car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 6.0 m/sm/s , they then have the same kinetic energy. What were the original speeds of the two cars? Express your answers using two significant figures separated by a comma.

Answer 1

Let's assume velocities are positive

Write out the equations:

The first mass is twice as much as the second mass so:

1) m₁ = 2m₂

The first mass has half the kinetic energy as the second mass so:

2 * (1/2)m₁v₁² = (1/2)m₂v₂²

Simplify:

2) 2 * m₁v₁² = m₂v₂²

When both cars increase their speed by 6m/s then they have the same kinetic energy so:

(1/2)m₁(v₁ + 6)² = (1/2)m₂(v₂ + 6)²

Simplify:

3) m₁(v₁ + 6)² = m₂(v₂ + 6)²

Replace m₁ with 2m₂ in equation 2

2 * 2m₂v₁² = m₂v₂²

4v₁² = v₂²

2v₁ = v₂

Now in equation 3, replace m₁ with 2m₂ and replace v₂ with 2v₁

2m₂(v₁ + 6)² = m₂(2v₁ + 6)²

2(v₁ + 6)² = (2v₁ + 6)²

2v₁² + 24v₁ + 72 = 4v₁² + 24v₁ + 36

2v₁² + 72 = 4v₁² + 36

2v₁² = 36

v₁² = 18

v₁ = 3√2

v₂ = 2v₁ = 6√2