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.

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

We know that m_{1 =} 2m₂ (1)

K_1i = 1/2 K_2i (2)

K_1f = K_2f (3)

Change of the velocity of both cars

Δv = 6.0 m/s.

Initial kinetic energy of the first car K_1i = 1/2 m₁v₁² .

Initial kinetic energy of the second car K_2i = 1/2 m₂v₂² .

Final kinetic energy of the first car K_1f= 1/2 m₁(v₁ + Δv)²

and final kinetic energy of the second car K_2f = 1/2 m₂(v₂ + Δv)²

Using (1) and (2), we can write: K_1i = 1/2 (2m₂)v₁² (4)

2K_1i = 1/2 m₂v₂² (5)

Dividing (4) by (5) we get 1/2 = 2v₁²/v₂²

or v₂2 = 4v₁2.

Hence, v₂ = 2v₁ ( ∗ )

Using (3) we write 1/2 m₁(v₁ + Δv)² = 1/2 m₂(v₂ + Δv)²

Knowing (1) we rewrite this as 2m₂(v₁ + Δv)² = m₂(v₂ + Δv)²

Or [(v₂ + Δv)²]/[(v₁ + Δv)²] = 2 .

Taking square root and knowing ( ∗ ) we simplify to

(2v₁ + Δv)/(v₁ + Δv) = √2

Putting value of Δv = 6 and solving for v₁ we obtain

v₁ = [6(√2 - 1)]/(2 - √2)

Then because of ( ∗ )

v₂ = [12(√2 - 1)]/(2 - √2)

Now it is easy to put in these equations the value of square root of 2 and get final numeric answer.

We can see that this is not physics, but mainly math problem. As famous Soviet physicist, Nobel Prize winner Lev Landau used to say:" You can be a physicist knowing math and not understand physics, but you cannot be a physicist without knowing math."

Good luck!