91Ó°ÊÓ

A toy train is moving along a straight length of track. It accelerates uniformly from rest to a velocity of \(0.5 \mathrm{~ms}^{-1}\) and maintains this velocity for a time before decelerating uniformly to rest again. If the time taken for this journey is \(2 \mathrm{sec}\) onds and it moves a distance of \(0.8 \mathrm{~m}\) along the track, find the time for which the speed of the train is uniform.

A particle P which is moving along a straight line with a constant acceleration of \(0.3 \mathrm{~ms}^{-2}\) passes a point \(\mathrm{A}\) on the line with a velocity of \(20 \mathrm{~ms}^{-1}\). At the time when P passes A a second particle \(Q\) is \(20 \mathrm{~m}\) behind \(A\) and is moving with a constant velocity of \(30 \mathrm{~ms}^{-1}\). Prove that the particles collide.

In a motor race, a car \(\mathrm{A}\) is \(1 \mathrm{~km}\) from the finishing post, and is travelling at \(35 \mathrm{~m}\) per second with a uniform acceleration of \(\frac{3}{3} \mathrm{~m}\) per \(\mathrm{sec}^{2}\). At the same instant a second car \(\mathrm{B}\) is \(200 \mathrm{~m}\) behind \(\mathrm{A}\) and is travelling at \(44 \mathrm{~m}\) per second with a uniform acceleration of \(\frac{1}{2} \mathrm{~m}\) per \(\sec ^{2} .\) Show that B passes A \(220 \mathrm{~m}\) before the finish. Show also that, if these accelerations are maintained, B arrives at the finishing. post \(1 \mathrm{sec}\). before \(\mathrm{A}\). (Cambridge)