3.02h Motion under gravity: vector form

6 A particle \(A\) is projected vertically upwards from level ground with an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(B\) is released from rest 15 m vertically above \(A\). The mass of one of the particles is twice the mass of the other particle. During the subsequent motion \(A\) and \(B\) collide and coalesce to form particle \(C\). Find the difference between the two possible times at which \(C\) hits the ground. \(7 \quad\) A particle \(P\) moving in a straight line starts from rest at a point \(O\) and comes to rest 16 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) is given by $$\begin{array} { l l } a = 6 + 4 t & 0 \leqslant t < 2 , \\ a = 14 & 2 \leqslant t < 4 , \\ a = 16 - 2 t & 4 \leqslant t \leqslant 16 . \end{array}$$ There is no sudden change in velocity at any instant.

1. Find the values of \(t\) when the velocity of \(P\) is \(55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Complete the sketch of the velocity-time diagram. \includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-11_511_1054_351_584}
3. Find the distance travelled by \(P\) when it is decelerating.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m .

1. Show that \(u = 22\).
2. The height of the particle above the ground is more than \(h \mathrm {~m}\) for a period of 3.6 s . Find \(h\).

2 A particle \(P\) is projected vertically upwards from horizontal ground. \(P\) reaches a maximum height of 45 m . After reaching the ground, \(P\) comes to rest without rebounding.

1. Find the speed at which \(P\) was projected.
2. Find the total time for which the speed of \(P\) is at least \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) reaches a maximum height of 20 m above the ground.

1. Find the value of \(u\).
2. Find the total time for which \(P\) is at least 15 m above the ground.

5 A particle is projected vertically upwards with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) alongside a building of height \(h \mathrm {~m}\).

1. Given that the particle is above the level of the top of the building for 4 s , find \(h\).
2. One second after the first particle is projected, a second particle is projected vertically upwards from the top of the building with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Denoting the time after projection of the first particle by \(t \mathrm {~s}\), find the value of \(t\) for which the two particles are at the same height above the ground.

1 A particle \(P\) is projected vertically upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(v\).
2. Find the greatest height of \(P\) above the ground.

3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.

1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.

1 A particle \(P\) is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(u\).
2. Find the greatest height of \(P\) above the ground.

1 A particle is projected vertically upwards from horizontal ground with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle has height \(s \mathrm {~m}\) above the ground at times 3 seconds and 4 seconds after projection. Find the value of \(u\) and the value of \(s\).

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.

1 A particle \(P\) is projected vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground.

1. Find the greatest height above the ground reached by \(P\).
2. Find the total time from projection until \(P\) returns to the ground.

7 A particle \(P _ { 1 }\) is projected vertically upwards, from horizontal ground, with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle \(P _ { 2 }\) is projected vertically upwards from the top of a tower of height 25 m , with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the time for which \(P _ { 1 }\) is higher than the top of the tower,
2. the velocities of the particles at the instant when the particles are at the same height,
3. the time for which \(P _ { 1 }\) is higher than \(P _ { 2 }\) and is moving upwards.

4 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_568_1084_269_532} The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point \(A\) above the surface of liquid in a container. The downward velocity of the stone \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone hits the surface of the liquid with velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0.7\). It reaches the bottom of the container with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 1.2\).

1. Find
   1. the height of \(A\) above the surface of the liquid,
   2. the depth of liquid in the container.
   3. Find the deceleration of the stone while it is moving in the liquid.
   4. Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N , find the mass of the stone.

7 Two particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. The particles start at the same instant and from the same point, each with speed \(1.3 \mathrm {~ms} ^ { - 1 }\). Initially \(P\) moves down the plane and \(Q\) moves up the plane. The distance between the particles \(t\) seconds after they start to move is \(d \mathrm {~m}\).

1. Show that \(d = 2.6 t\). When \(t = 2.5\) the difference in the vertical height of the particles is 1.6 m . Find
2. the acceleration of the particles down the plane,
3. the distance travelled by \(P\) when \(Q\) is at its highest point.

6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find

1. the greatest height above the ground reached by \(P\),
2. the speed with which \(P\) reaches the ground,
3. the total work done on \(P\) by the air resistance.

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 \(t\).

5 Two particles \(P\) and \(Q\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(P\) and \(Q\) are \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively and the heights of \(P\) and \(Q\) above the ground, \(t\) seconds after projection, are \(h _ { P } \mathrm {~m}\) and \(h _ { Q } \mathrm {~m}\) respectively. Each particle comes to rest on returning to the ground.

1. Find the set of values of \(t\) for which the particles are travelling in opposite directions.
2. At a certain instant, \(P\) and \(Q\) are above the ground and \(3 h _ { P } = 8 h _ { Q }\). Find the velocities of \(P\) and \(Q\) at this instant.

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find

1. the acceleration of \(B\) while it is moving downwards,
2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground. \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.

3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find

1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
3. Find the speed of projection of the second signal.

5 A particle \(P\) is projected vertically upwards from a point on the ground with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) is projected vertically upwards from the same point with speed \(7 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is projected \(T\) seconds later than particle \(P\).

1. Given that the particles reach the ground at the same instant, find the value of \(T\).
2. At a certain instant when both \(P\) and \(Q\) are in motion, \(P\) is 5 m higher than \(Q\). Find the magnitude and direction of the velocity of each of the particles at this instant.

4 A small ball of mass 0.4 kg is released from rest at a point 5 m above horizontal ground. At the instant the ball hits the ground it loses 12.8 J of kinetic energy and starts to move upwards.

1. Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
2. Find the time taken for the ball's motion from its release until reaching this greatest height.

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.

5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.

1. Find \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
2. Hence find the height above \(O\) at which the particles collide.

2 A particle \(P\) is projected vertically upwards with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 3 m above horizontal ground.

1. Find the time taken for \(P\) to reach its greatest height.
2. Find the length of time for which \(P\) is higher than 23 m above the ground.
3. \(P\) is higher than \(h \mathrm {~m}\) above the ground for 1 second. Find \(h\).