2024 CGSS P2 Q9

Mr Chan was driving a car along a straight road. He was \(35\) m away from the stop line when he applied his brakes near a traffic light. His acceleration, \(a\) \(\text{m/s}^2\), after applying the brakes was given by \(a = -7.5\mathrm{e}^{-\frac{t}{2}}\), where \(t\) is the time in seconds after he applied the brakes.

1. Explain mathematically why \(a < 0\) for all \(t \ge 0\) and the significance of \(a < 0\).[2]
2. Mr Chan’s car was travelling at \(14\) m/s just before he applied his brakes.
   Express the velocity of his car, \(\mathrm{v}\) m/s, in terms of \(t\).[3]
3. Hence find the time taken for his car to come to a complete stop.[2]
4. Obtain an expression, in terms of \(t\), for the displacement of Mr Chan’s car from the point he applied the brakes.[3]
5. Determine if Mr Chan’s car was able to come to a complete stop before reaching the stop line. Explain your answer.[2]