Displacement expressions and comparison

6 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_593_746_269_701} A particle \(P\) starts from rest at the point \(A\) and travels in a straight line, coming to rest again after 10 s . The velocity-time graph for \(P\) consists of two straight line segments (see diagram). A particle \(Q\) starts from rest at \(A\) at the same instant as \(P\) and travels along the same straight line as \(P\). The velocity of \(Q\) is given by \(v = 3 t - 0.3 t ^ { 2 }\) for \(0 \leqslant t \leqslant 10\). The displacements from \(A\) of \(P\) and \(Q\) are the same when \(t = 10\).

1. Show that the greatest velocity of \(P\) during its motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(t\), in the interval \(0 < t < 5\), for which the acceleration of \(Q\) is the same as the acceleration of \(P\).

6 Particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. \(P\) is released from rest at a point \(O\) on the line and 2 s later passes through the point \(A\) with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) and the angle of inclination of the plane. At the instant that \(P\) passes through \(A\) the particle \(Q\) is released from rest at \(O\). At time \(t\) s after \(Q\) is released from \(O\), the particles \(P\) and \(Q\) are 4.9 m apart.
2. Find the value of \(t\).

2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.

1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
2. Find the distance \(O P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_240_862_274_644} Particles \(P\) and \(Q\) start from points \(A\) and \(B\) respectively, at the same instant, and move towards each other in a horizontal straight line. The initial speeds of \(P\) and \(Q\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The accelerations of \(P\) and \(Q\) are constant and equal to \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively (see diagram).

1. Find the speed of \(P\) at the instant when the speed of \(P\) is 1.8 times the speed of \(Q\).
2. Given that \(A B = 51 \mathrm {~m}\), find the time taken from the start until \(P\) and \(Q\) meet.

4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).

5 Particle \(P\) travels along a straight line from \(A\) to \(B\) with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Its speed at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time taken for \(P\) to travel from \(A\) to \(B\), and find also the distance \(A B\). Particle \(Q\) also travels along the same straight line from \(A\) to \(B\), starting from rest at \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the speed of \(Q\) is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. \(Q\) takes the same time to travel from \(A\) to \(B\) as \(P\) does.
2. Find the value of \(k\) and find \(Q\) 's speed at \(B\).

3 A man travels 360 m along a straight road. He walks for the first 120 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), runs the next 180 m at \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then walks the final 60 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \(( t , x )\) graph for the journey, showing the values of \(t\) for which \(x = 120,300\) and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
2. Draw a dotted line on your ( \(t , x\) ) graph to represent the woman's journey.
3. Calculate the value of \(t\) at which the man overtakes the woman.

3 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start

• P is travelling directly towards Q with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• Q is instantaneously at rest and has an acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directly towards P . \(P\) continues with the same velocity and \(Q\) continues with the same acceleration. The cars collide \(T\) seconds after the start.
  1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start.

At the start, \(P\) is 90 m from \(Q\).

Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).

4 \begin{figure}[h] \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards Q of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle Q starts 125 m from P at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from P . The initial values are shown in Fig. 4.

1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
2. How much time does it take for P to catch up with Q and how far does P travel in this time?

3 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by $$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$ Find the values of \(x\) for which the velocity of the particle is zero.

Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u\) m s\(^{-1}\) Jermaine is walking 0.2 m s\(^{-1}\) more slowly than Meena. When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of 2 m s\(^{-2}\), for a time of \(t\) seconds, until he reaches her.

1. Show that $$d = t^2 - 0.2t$$ [4 marks]
2. When Jermaine's speed is 7.8 m s\(^{-1}\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\). [2 marks]

The displacement, \(r\) metres, of a particle at time \(t\) seconds is $$r = 6t - 2t^2$$

1. Find the value of \(r\) when \(t = 4\) [1 mark]
2. Determine the range of values of \(t\) for which the displacement is positive. [2 marks]