2026 NYGH P2 Q14

A robot is set to move on a straight track towards a wall during the testing of its automated braking system. When the robot's proximity sensors detect a wall, it triggers the braking system, which causes it to slow down. Its velocity \(v\text{ m/s}\), at \(t\) seconds, after the braking system is triggered, is given by \(v=15\mathrm{e}^{-0.5t}-1\).

1. Show that the initial velocity is \(14\text{ m/s}\).[1]
2. Explain why the robot’s acceleration is always negative.[2]
3. Calculate the time taken for the robot to come to a complete stop.[3]
4. Find an expression, in terms of \(t\), for the displacement of the robot after the braking system is triggered.[3]

A data-logger records that the robot is exactly \(35\text{ m}\) from the wall at the instant the braking system is triggered.

5. Determine, by showing your working clearly, whether the robot is able to come to a complete stop before hitting the wall.[2]