2025 Nov TZ1 P2 Q9

An airplane lands on a runway \(100\) metres in front of a stationary car. At the instant the airplane lands, the car begins to travel in the same direction towards the airplane.

Let \(t\) represent the number of seconds after the airplane lands. For \(t \geq 0\), the velocities of the airplane and the car in m s\(^{-1}\), can be modelled by the following equations:

\(\mathrm{v}_{air} = 60 \mathrm{e}^{-0.1t}\)
\(\mathrm{v}_{car} = 5t \)

1. When the airplane lands, write down the speed of

   1. the airplane;

   2. the car.

      [2]

2. Find

   1. the value of \(t\) when the airplane and the car have the same speed;

   2. the speed at this time.

      [3]

Let \(d(t)\) represent the distance, in metres, in between the car and the back of the airplane after \(t\) seconds.

3. If \(d(0) = 100\), find \(d(t)\).

   [7]

4. Hence, find how long it takes for the car to reach the back of the airplane.

   [2]

5. Find the distance travelled by the car when it reaches the back of the airplane.

   [3]