Chapter 4

A stone is dropped from the top of a building \(20 \mathrm{~m}\) high. A second stone is dropped from half-way up the same building. Find the time that should elapse between the release of the two stones if they are to reach the ground at the same time.

A particle is describing a circle of radius \(4 \mathrm{~m}\) with a constant angular accel. eration. At one instant it has a speed of \(2 \mathrm{~ms}^{-1}\) and \(4 \mathrm{~s}\) later it has a speed of \(10 \mathrm{~ms}^{-1}\). Find its angular acceleration and the distance it has travelled in this time.

A particle is describing a vertical circle of radius \(2 \mathrm{~m}\) with a constant angular acceleration of \(\frac{\pi}{6} \mathrm{rad} \mathrm{s}^{-2}\). If it is initially at rest at the lowest point of the circle find its speed 2 seconds later and its displacement from its original position.

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 moving in a straight line with a constant acceleration of \(3 \mathrm{~ms}^{-2}\) has an initial velocity of \(-1 \mathrm{~ms}^{-1}\). Its velocity two seconds later is: (a) \(5 \mathrm{~ms}^{-1}\) (b) \(6 \mathrm{~ms}^{-1}\) (c) \(4 \mathrm{~ms}^{-1}\) (d) 0 (e) \(-7 \mathrm{~ms}^{-1}\)

A car has a maximum acceleration of \(6 \mathrm{~ms}^{-2}\) and a maximum deceleration of \(8 \mathrm{~ms}^{-2}\). Find the least time in which it can cover a distance of \(0.2 \mathrm{~km}\) starting from rest and stopping again. What is the maximum speed reached by the car in this time?

A particle moves in a straight line and passes through \(\mathrm{O}\), a fixed point on the line with a velocity of \(6 \mathrm{~ms}^{-1}\). The particle moves with a constant retardation of \(2 \mathrm{~ms}^{-2}\) for four seconds and thereafter moves with constant velocity. How long after leaving \(\mathrm{O}\) does the particle return to \(\mathrm{O}\) ? (a) \(3 \mathrm{~s}\) (b) \(8 \mathrm{~s}\) (c) never (d) \(4 \mathrm{~s}\) (e) \(6 \mathrm{~s}\).

A particle P moves along the \(x\)-axis and a particle \(Q\) moves along the \(y\)-axis. P starts from rest at the origin and moves with a constant acceleration of \(2 \mathrm{~ms}^{-2}\). At the same time \(Q\) is at the point \((0,3)\) with a velocity of \(2 \mathrm{~ms}^{-1}\) and is moving with a constant acceleration of \(-3 \mathrm{~ms}^{-2} .\) Find the distance between \(\mathrm{P}\) and \(Q\) 4 seconds later.

If a particle is moving in a straight line with constant acceleration and a velocity-time graph is drawn for the motion, the gradient of the graph represents: (a) the acceleration, (b) the rate of increase of velocity, (c) the rate of decrease of velocity.

A particle P starts from rest from a point \(A\) and moves along a straight line with a constant acceleration of \(2 \mathrm{~ms}^{-2}\). At the same time a second particle \(Q\) is \(5 \mathrm{~m}\) behind \(\mathrm{A}\) and is moving in the same direction as \(\mathrm{P}\) with a speed of \(5 \mathrm{~ms}^{-1}\). If \(Q\) has a constant acceleration \(3 \mathrm{~ms}^{-2}\) find how far from \(A\) it overtakes \(P\).

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.

(a) A particle is moving in a straight line with constant acceleration. (b) The average velocity of a particle moving in a straight line is the algebraic average of the initial and final velocities.

A bus moves away from rest at a bus stop with an acceleration of \(1 \mathrm{~ms}^{-2} .\) As the bus starts to move a man who is \(4 \mathrm{~m}\) behind the stop runs with a constant speed after the bus. If he justs manages to catch the bus find his speed.

(a) A particle is moving in a straight line with a constant acceleration of \(2 \mathrm{~ms}^{-2}\) (b) A particle moving in a straight line with a constant acceleration has a velocity of \(2 \mathrm{~ms}^{-1}\) at one instant and a veloctty of \(8 \mathrm{~ms}^{-1}\) three seconds later.

A model aeroplane is constrained to fly in a circle by a guide line which is \(3 \mathrm{~m}\) long. It accelerates from a speed of \(2 \mathrm{~ms}^{-1}\) with a constant angular acceleration of \(\frac{\pi}{10} \mathrm{rad} \mathrm{s}^{-2}\) for \(2 \frac{1}{2}\) revolutions. The guide line then breaks. Find the speed of the aeroplane when the guide line breaks.

A stone is thrown vertically upward with a speed of \(u\) metre per second. \(A\) second stone is thrown vertically upward from the same point with the same initial speed but \(T\) seconds later than the first one. Prove that they collide at a distance of \(\left(\frac{4 u^{2}-g^{2} T^{2}}{8 g}\right)\) metre above the point of projection.

A particle moving in a straight line OD with uniform retardation leaves point \(\mathrm{O}\) at time \(t=0\), and comes to instantaneous rest at D. On its way to D the particle passes points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) at times \(t=T, 2 T, 4 T\), respectively after leaving \(\mathrm{O}\), where \(\mathrm{AB}=\mathrm{BC}=l\). Find, in terms of \(l\), (i) the length of \(\mathrm{CD}\) and (ii) the length of \(\mathrm{OA} .\) (J.M.B.)

15) Three points A, B, C on a motor racing track are such that \(B\) is \(1 \mathrm{~km}\) beyond A and \(\mathrm{C}\) is \(2 \mathrm{~km}\) beyond B. A car \(\mathrm{X}\), moving with uniform acceleration takes 1 minute to travel from \(A\) to \(B\) and \(1 \frac{1}{2}\) minutes to travel from \(B\) to \(C\). Find its acceleration in \(\mathrm{km} / \mathrm{h} / \mathrm{min}\) and show that its speed at \(\mathrm{C}\) is \(92 \mathrm{~km} / \mathrm{h}\). Another car Y, which is moving with uniform acceleration of \(8 \mathrm{~km} / \mathrm{h}\) per min. passes C 15 . seconds earlier than \(\mathrm{X}\), and its speed is then \(75 \mathrm{~km} / \mathrm{h}\). Find where \(\mathrm{X}\) passes \(\mathrm{Y}\). (Cambridge)

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)

A particle moving in a straight line with a constant acceleration of \(-2 \mathrm{~ms}^{-2}\) has a velocity of \(3 \mathrm{~ms}^{-1}\) at one instant and a velocity of \(-3 \mathrm{~ms}^{-1}\) three seconds later.

A flywheel starts from rest and is uniformly accelerated to an angular speed of 120 revolutions per minute. It maintains this speed until it is uniformly retarded to rest again. The magnitude of the retardation is three times the value of the starting acceleration. Between starting and coming to rest again the flywheel completes \(N\) revolutions in five minutes. Sketch the angular speed-time graph and hence find, in terms of \(N\), the time for which the flywheel is travelling at the maximum speed. Show that \(300

A particle is moving in the positive sense on the circumference of a circle of radius \(2 \mathrm{~m}\). The particle has a constant angular acceleration of \(3 \mathrm{rad} \mathrm{s}^{-2}\). At one instant the speed of the particle is \(2 \mathrm{~ms}^{-1}\) and one second later it is \(8 \mathrm{~ms}^{-1}\).

A particle is moving in a straight line. A displacement-time graph is drawn for its motion. The gradient of the tangent to the graph at time \(T\) represents the. velocity of the particle at time \(T\).