Calculating the Time for Cars to Overtake Each Other

A car starts from rest and accelerates uniformly at 3 m/s^2 towards north. A second car starts from rest.

Since the second car starts 6.0 s later than the first car, the time it takes for the second car to overtake the first car is 12.0 s Option A

How to calculate the times for the car

To determine the time it takes for the second car to overtake the first car, use the equations of motion.

Let's denote the time it takes for the second car to catch up to the first car as t.

For the first car:

Initial velocity (u₁) = 0 (since it starts from rest)

Acceleration (a₁) = 3.0 m/s²

Time (t) = t

For the second car:

Initial velocity (u₂) = 0 (since it starts from rest)

Acceleration (a₂) = 5.0 m/s²

Time (t) = t - 6.0 s (since it starts 6.0 s later)

Use the equation of motion to find the displacement (s) of each car at time t:

For the first car:

s₁ = u₁t + (1/2)a₁t²

For the second car:

s₂ = u₂(t - 6.0) + (1/2)a₂(t - 6.0)²

Since the second car overtakes the first car, their displacements will be equal:

s₁ = s₂

From the above equations, equate the expressions for s₁ and s₂:

u₁t + (1/2)a₁t² = u₂(t - 6.0) + (1/2)a₂(t - 6.0)²

Substitute the given values:

0 + (1/2)(3.0)(t)² = 0 + (1/2)(5.0)(t - 6.0)²

1.5t² = 2.5(t - 6.0)²

Expand and rearrange:

1.5t² = 2.5(t² - 12t + 36)

1.5t² = 2.5t² - 30t + 90

1.5t² - 2.5t² + 30t - 90 = 0

t² - 20t + 60 = 0

By solving this quadratic equation using the quadratic formula,

t = 6 or t = 10

Since the second car starts 6.0 s later than the first car, the time it takes for the second car to overtake the first car is:

t = 6.0 s + 6.0 s = 12.0 s