Physics problem, find acceleration

Question

A car is dragging an empty trailer with a1=0.4 m/s^2, a full trailer with a2=0.1 m/s^2. Find acceleration a3 when the car is dragging an empty and a full trailers together?

The correct answer is 0.08 m/s^2

Answer 1

Assume the force exerted by the car is the same in all three trials, and use Newton's Second Law (F = ma).

Comparing the empty trailer to the full one, F₁ = F₂, so m₁a₁ = m₂a₂. As a₁ = 4*a₂, we can write the following:

m₁(4 a₂) = m₂a₂

4 m₁ = m₂, or m₂ = 4 m₁ (full has 4x mass of empty)

When the car tows both trailers, the combined mass equal the masses of the empty and the full. Therefore, we can say m₃ = m₁ + m₂. With m₂ = 4 m₁ , m₃ = m₁ + 4 m₁ = 5m₁.

Applying the same force to five times the mass yields one-fifth of the original acceleration (a₃ = 1/5 a₁).

So the acceleration with both trailers equals (1/5) * 0.4 m/s², which is 0.08 m/s².