Calculating Time for Mate to Catch up with Bus

Calculation:

Given data:

Uniform acceleration of the bus: 2m/s²

Starting position of the mate behind the bus: 9m

Uniform speed of the mate: 6m/s

Let's denote the position of the bus as x_bus at time t:

x_bus = (1/2) * (2m/s²) * t²

If we take the bus's rear's starting position to be the origin, then the position of the mate, x_mate, is:

x_mate = -9m + (6m/s) * t

The mate catches up with the bus when x_bus = x_mate:

(1/2) * (2m/s²) * t² = -9m + (6m/s) * t

Simplifying the equation:

2t² = -9 + 6t

t² - 6t + 9 = (t - 3)² = 0

Therefore, t = 3 seconds

Thus, the mate will take 3 seconds to catch up with the bus.