Question

One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 9.0 m/s , they then have the same kinetic energy.

a)What were the original speeds of the two cars?

Answer 1

Mass of the first car = m₁

Mass of the second car = m₂

Mass of the first car is twice that of the second car.

m₁ = 2m₂

Initial speed of the first car = V₁

Initial speed of the second car = V₂

Initial kinetic energy of the first car = KE₁ = m₁V₁²/2

Initial kinetic energy of the second car = KE₂ = m₂V₂²/2

The initial kinetic energy of the first car is half of that of the second car.

KE₁ = 0.5KE₂

m₁V₁²/2 = 0.5(m₂V₂²/2)

m₁V₁² = 0.5m₂V₂²

(2m₂)V₁² = 0.5m₂V₂²

4V₁² = V₂²

2V₁ = V₂

Now both cars increase their speed by 9 m/s

New speed of the first car = V₃ = V₁ + 9

New speed of the second car = V₄ = V₂ + 9

New kinetic energy of the first car = KE₃ = m₁V₃²/2

New kinetic energy of the second car = KE₄ = m₂V₄²/2

Now the kinetic energies of both the cars are equal.

KE₃ = KE₄

m₁V₃²/2 = m₂V₄²/2

m₁V₃² = m₂V₄²

(2m₂)V₃² = m₂V₄²

2V₃² = V₄²

1.414V₃ = V₄

1.414(V₁ + 9) = V₂ + 9

1.414V₁ + 12.726 = 2V₁ + 9

0.586V₁ = 3.726

V₁ = 6.36 m/s

V₂ = 2V₁

V₂ = 2(6.36)

V₂ = 12.72 m/s

a) Original speed of the first car = 6.36 m/s

b) Original speed of the second car = 12.72 m/s