12.
Answer all the questions.
Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track.
The cyclists are modelled as particles and the motion of the cyclists is modelled as follows:
- At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\)
- Cyclist \(A\) is moving in a straight line with constant acceleration \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
- At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\)
- Cyclist \(B\) moves with constant acceleration \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) along the same straight line and in the same direction as cyclist \(A\)
- At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\)
Using the model,
- sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds,
- a velocity-time graph for the motion of \(A\)
- a velocity-time graph for the motion of \(B\)
- explain why the two graphs must cross before time \(t = T\) seconds,
- find the time when \(A\) and \(B\) are moving at the same speed,
- find the distance \(O X\)
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