5. A car travels at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) in a straight line along a horizontal racetrack. At time \(t = 0\), the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car.
The motorcyclist accelerates at \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 15 s and then accelerates at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a further \(T\) seconds until he catches up with the car.
- Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time \(t = 0\) to the instant when the motorcyclist catches up with the car.
At the instant when \(t = t _ { 1 }\) seconds, the car and the motorcyclist are moving at the same speed.
- Find the value of \(t _ { 1 }\)
- Show that \(T ^ { 2 } + k T - 300 = 0\), where \(k\) is a constant to be found.
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