Behind the start despite moving forward
Behind the start despite moving forward
IP 4
9 marks
A particle moves in a straight line so that its velocity \(t\) seconds after passing a fixed point \(O\) is \(v=2\cos t-1\) m/s, for \(0\le t\le2\pi\).
- Find the exact times at which the particle is at rest.[2]
- Show that the displacement of the particle from \(O\) when \(t=2\pi\) is \(-2\pi\) m, and explain why it ends behind \(O\) even though it moves in the positive direction at the start.[3]
- Find the exact total distance travelled for \(0\le t\le2\pi\).[4]
Solution:
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Answer:(a) \(t=\dfrac{\pi}{3}\) and \(t=\dfrac{5\pi}{3}\) (b) \(s=2\sin t-t\), \(s(2\pi)=-2\pi\); the velocity is negative for most of the interval (c) \(4\sqrt3+\dfrac{2\pi}{3}\approx9.02\) m
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