A car travels at a constant speed of $$\(40 \mathrm{~m} \mathrm{~s}^{-1}\)$$ in a straight line along a horizontal racetrack. At time $$\(t=0\)$$, the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car. The motorcyclist accelerates at $$\(4 \mathrm{~m} \mathrm{~s}^{-2}\)$$ for $$\(15 \mathrm{~s}\)$$ and then accelerates at $$\(1 \mathrm{~m} \mathrm{~s}^{-2}\)$$ for a further $$\(T\)$$ seconds until he catches up with the car. (a) Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time $$\(t=0\)$$ to the instant when the motorcyclist catches up with the car. (2) At the instant when $$\(t=t_{1}\)$$ seconds, the car and the motorcyclist are moving at the same speed. (b) Find the value of $$\(t_{1}\)$$ (2) (c) Show that $$\(T^{2}+k T-300=0\)$$, where $$\(k\)$$ is a constant to be found. (6)

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