2. A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) is given by
$$a = \begin{cases} 4 t - t ^ { 2 } , & 0 \leq t \leq 3 ,
\frac { 27 } { t ^ { 2 } } , & t > 3 . \end{cases}$$
At \(t = 0 , P\) is at rest. Find the speed of \(P\) when
- \(t = 3\),
- \(t = 6\).
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m . When she reaches \(B\), her speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m . By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N , - find the work done by the cyclist in moving from \(A\) to \(B\).
At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
- find the power generated by the cyclist at \(B\).