3.02h Motion under gravity: vector form

4 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find

1. the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height,
2. the height of \(A\) above the ground when \(B\) is 0.9 m above \(A\).

2 A stone is released from rest and falls freely under gravity. Find

1. the speed of the stone after 2 s ,
2. the time taken for the stone to fall a distance of 45 m from its initial position,
3. the distance fallen by the stone from the instant when its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to the instant when its speed is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.

1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
2. At the instant that the string breaks, find
   1. the height above the ground of \(P\) and of \(Q\),
   2. the speed of the particles.
   3. Show that \(Q\) reaches the ground 0.8 s later than \(P\). \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
      1. the speed of \(P\) when \(t = 10\), and the value of \(a\),
      2. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
      3. the displacement of \(P\) from \(A\) when \(t = 20\).

1 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_608_831_258_657} Two particles \(P\) and \(Q\) move vertically under gravity. The graphs show the upward velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particles at time \(t \mathrm {~s}\), for \(0 \leqslant t \leqslant 4 . P\) starts with velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) starts from rest.

1. Find the value of \(V\). Given that \(Q\) reaches the horizontal ground when \(t = 4\), find
2. the speed with which \(Q\) reaches the ground,
3. the height of \(Q\) above the ground when \(t = 0\).

2 A cyclist, working at a constant rate of 400 W , travels along a straight road which is inclined at \(2 ^ { \circ }\) to the horizontal. The total mass of the cyclist and his cycle is 80 kg . Ignoring any resistance to motion, find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling

1. uphill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. downhill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 Particles \(P\) and \(Q\) are projected vertically upwards, from different points on horizontal ground, with velocities of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(Q\) is projected 0.4 s later than \(P\). Find

1. the time for which \(P\) 's height above the ground is greater than 15 m ,
2. the velocities of \(P\) and \(Q\) at the instant when the particles are at the same height.

1 A particle \(P\) is released from rest at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Find the speed of \(P\)

1. when it has travelled 0.9 m ,
2. 0.8 s after it is released.

4 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438} \(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find

1. the values of \(u\) and \(a\),
2. the value of \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.

1 An object is released from rest at a height of 125 m above horizontal ground and falls freely under gravity, hitting a moving target \(P\). The target \(P\) is moving on the ground in a straight line, with constant acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant the object is released \(P\) passes through a point \(O\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance from \(O\) to the point where \(P\) is hit by the object.

3 A particle \(P\) is projected vertically upwards, from a point \(O\), with a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is the highest point reached by \(P\). Find

1. the speed of \(P\) when it is at the mid-point of \(O A\),
2. the time taken for \(P\) to reach the mid-point of \(O A\) while moving upwards.

6 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor. \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).

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 \(V\).

2 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_262_1004_760_575} The tops of each of two smooth inclined planes \(A\) and \(B\) meet at a right angle. Plane \(A\) is inclined at angle \(\alpha\) to the horizontal and plane \(B\) is inclined at angle \(\beta\) to the horizontal, where \(\sin \alpha = \frac { 63 } { 65 }\) and \(\sin \beta = \frac { 16 } { 65 }\). A small smooth pulley is fixed at the top of the planes and a light inextensible string passes over the pulley. Two particles \(P\) and \(Q\), each of mass 0.65 kg , are attached to the string, one at each end. Particle \(Q\) is held at rest at a point of the same line of greatest slope of the plane \(B\) as the pulley. Particle \(P\) rests freely below the pulley in contact with plane \(A\) (see diagram). Particle \(Q\) is released and the particles start to move with the string taut. Find the tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_487_696_1537_721} Each of three light inextensible strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(O\). Two of the strings pass over fixed smooth pegs and the particles hang freely in equilibrium. The weights of the particles and the angles between the sloping parts of the strings and the vertical are as shown in the diagram. It is given that \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that \(W \cos \alpha = 3.8\) and find the value of \(W \sin \alpha\).
2. Hence find the values of \(W\) and \(\alpha\).

2 A particle is released from rest at a point \(H \mathrm {~m}\) above horizontal ground and falls vertically. The particle passes through a point 35 m above the ground with a speed of \(( V - 10 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and reaches the ground with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(V\),
2. the value of \(H\).

3 A particle \(P\) is projected vertically upwards from a point \(O\). When the particle is at a height of 0.5 m , its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the greatest height reached by the particle above \(O\),
2. the time after projection at which the particle returns to \(O\).

4 A ball \(A\) is released from rest at the top of a tall tower. One second later, another ball \(B\) is projected vertically upwards from ground level near the bottom of the tower with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two balls are at the same height 1.5 s after ball \(B\) is projected.

1. Show that the height of the tower is 50 m .
2. Find the length of time for which ball \(B\) has been in motion when ball \(A\) reaches the ground. Hence find the total distance travelled by ball \(B\) up to the instant when ball \(A\) reaches the ground.

7 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.

1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
   At the instant when \(B\) reaches the ground, the string breaks.
2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground, in a direction making an angle of \(\alpha ^ { \circ }\) above the horizontal. The particle reaches the ground again after 12 s . Find

1. the value of \(\alpha\),
2. the greatest height reached by the particle,
3. the length of time for which the direction of motion of the particle is between \(20 ^ { \circ }\) above the horizontal and \(20 ^ { \circ }\) below the horizontal,
4. the horizontal distance travelled by the particle in the time found in part (iii).

5 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-4_547_933_269_607} Particles \(A\) and \(B\) are projected simultaneously from the top \(T\) of a vertical tower, and move in the same vertical plane. \(T\) is 7.2 m above horizontal ground. \(A\) is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is projected at an angle of \(60 ^ { \circ }\) above the horizontal with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } . A\) and \(B\) move away from each other (see diagram).

1. Find the time taken for \(A\) to reach the ground. At the instant when \(A\) hits the ground,
2. show that \(B\) is approximately 5.2 m above the ground,
3. find the distance \(A B\).

5 A small stone is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. Referred to horizontal and vertically upwards axes through \(O\), the equation of the stone's trajectory is \(y = 0.75 x - 0.02 x ^ { 2 }\), where \(x\) and \(y\) are in metres. Find

1. the values of \(\theta\) and \(V\),
2. the distance from \(O\) of the point where the stone hits the ground,
3. the greatest height reached by the stone.

5 A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).

1. Using the equation of the particle's trajectory and the identity \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\), show that the possible values of \(\tan \theta\) are \(\frac { 3 } { 4 }\) and \(\frac { 17 } { 4 }\).
2. Find the distance \(O A\) for each of the two possible values of \(\tan \theta\).
3. Sketch in the same diagram the two possible trajectories.

7 The equation of the trajectory of a projectile is \(y = 0.6 x - 0.017 x ^ { 2 }\), referred to horizontal and vertically upward axes through the point of projection.

1. Find the angle of projection of the projectile, and show that the initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed and direction of motion of the projectile when it is at a height of 5.2 m above the level of the point of projection for the second time.

1 A small ball is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\).
2. Show that the equation of the trajectory of the ball is \(y = x - \frac { 1 } { 40 } x ^ { 2 }\).
3. State the distance from \(O\) of the point at which the ball first strikes the ground.

5 A particle \(P\) is projected with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point \(O\). For the instant 2.5 s after projection, calculate

1. the speed of \(P\),
2. the angle between \(O P\) and the horizontal.