Two particles: same start time, different heights

5 Two particles, \(P\) and \(Q\), of masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are held at rest in the same vertical line. The heights of \(P\) and \(Q\) above horizontal ground are 1 m and 2 m respectively. \(P\) is projected vertically upwards with speed \(2 \mathrm {~ms} ^ { - 1 }\). At the same instant, \(Q\) is released from rest.

1. Find the speed of each particle immediately before they collide.
2. It is given that immediately after the collision the downward speed of \(Q\) is \(3.5 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(P\) at the instant that it reaches the ground.

3 \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-2_443_825_755_661} A particle \(P\) is projected from the top of a smooth ramp with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and travels down a line of greatest slope. The ramp has length 6.4 m and is inclined at \(30 ^ { \circ }\) to the horizontal. Another particle \(Q\) is released from rest at a point 3.2 m vertically above the bottom of the ramp, at the same instant that \(P\) is projected (see diagram). Given that \(P\) and \(Q\) reach the bottom of the ramp simultaneously, find

1. the value of \(u\),
2. the speed with which \(P\) reaches the bottom of the ramp. \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-3_609_1539_255_303} The diagram shows the velocity-time graphs for the motion of two particles \(P\) and \(Q\), which travel in the same direction along a straight line. \(P\) and \(Q\) both start at the same point \(X\) on the line, but \(Q\) starts to move \(T\) s later than \(P\). Each particle moves with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first 20 s of its motion. The speed of each particle changes instantaneously to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after it has been moving for 20 s and the particle continues at this speed.

5 A particle \(P\) starts from rest at a point \(O\) on a horizontal straight line. \(P\) moves along the line with constant acceleration and reaches a point \(A\) on the line with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the instant that \(P\) leaves \(O\), a particle \(Q\) is projected vertically upwards from the point \(A\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(P\) and \(Q\) collide at \(A\). Find

1. the acceleration of \(P\),
2. the distance \(O A\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_538_414_315_370} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_561_686_264_1080} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Two particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively. The particles are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest with the string taut and both straight parts of the string vertical. \(P\) and \(Q\) are each at a height of \(h \mathrm {~m}\) above horizontal ground (see Fig. 1). \(P\) is released and \(Q\) moves downwards. Subsequently \(Q\) hits the ground and comes to rest. Fig. 2 shows the velocity-time graph for \(P\) while \(Q\) is moving downwards or is at rest on the ground.

1 A particle \(P\) is projected vertically upwards with speed \(11 \mathrm {~ms} ^ { - 1 }\) from a point on horizontal ground. At the same instant a particle \(Q\) is released from rest at a point \(h \mathrm {~m}\) above the ground. \(P\) and \(Q\) hit the ground at the same instant, when \(Q\) has speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time after projection at which \(P\) hits the ground.
2. Hence find the values of \(h\) and \(V\).

5 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614} \(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.

1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
   (b) Calculate \(\theta\).
   (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
2. Calculate the speeds of the particles immediately before they collide.

6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h] \end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). Section B (36 marks)

A particle \(P\) is projected vertically upwards from horizontal ground with speed \(15\,\text{m}\,\text{s}^{-1}\).

1. Find the speed of \(P\) when it is 10 m above the ground. [2] At the same instant that \(P\) is projected, a second particle \(Q\) is dropped from a height of 18 m above the ground in the same vertical line as \(P\).
2. Find the height above the ground at which the two particles collide. [3]

At time \(t = 0\), two balls \(A\) and \(B\) are projected vertically upwards. The ball \(A\) is projected vertically upwards with speed 2 m s\(^{-1}\) from a point 50 m above the horizontal ground. The ball \(B\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). At time \(t = T\) seconds, the two balls are at the same vertical height, \(h\) metres, above the ground. The balls are modelled as particles moving freely under gravity. Find

1. the value of \(T\), [5]
2. the value of \(h\). [2]

In this question use \(g = 9.8\) m s\(^{-2}\) An apple tree stands on horizontal ground. An apple hangs, at rest, from a branch of the tree. A second apple also hangs, at rest, from a different branch of the tree. The vertical distance between the two apples is \(d\) centimetres. At the same instant both apples begin to fall freely under gravity. The first apple hits the ground after 0.5 seconds. The second apple hits the ground 0.1 seconds later. Show that \(d\) is approximately 54 [4 marks]