3.02h Motion under gravity: vector form

2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(27 ^ { \circ }\).

4 A ball is projected from a point \(O\) on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. At time \(t\) seconds after projection the ball is at the point \(( x , y )\) referred to horizontal and vertically upward axes through \(O\). Air resistance may be neglected.

1. Express \(x\) and \(y\) in terms of \(t\), and hence show that \(y = 3 x - \frac { 1 } { 10 } x ^ { 2 }\). The ball hits the sea at a point which is 25 m below the level of \(O\).
2. Find the horizontal distance between the cliff and the point where the ball hits the sea.

4 A golfer hits a ball from a point \(O\) on horizontal ground with a velocity of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The horizontal range of the ball is \(R\) metres and the time of flight is \(t\) seconds.

1. Express \(t\) in terms of \(\theta\), and hence show that \(R = 125 \sin 2 \theta\). The golfer hits the ball so that it lands 110 m from \(O\).
2. Calculate the two possible values of \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_672_403_267_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A toy is constructed by attaching a small ball of mass 0.01 kg to one end of a uniform rod of length 10 cm whose other end is attached to the centre of the plane face of a uniform solid hemisphere with radius 3 cm . The rod has mass 0.02 kg , the hemisphere has mass 0.5 kg and the rod is perpendicular to the plane face of the hemisphere (see Fig. 1).

7 \includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.

7 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-4_440_657_906_744} A ball is projected with an initial speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) below the horizontal from a point on the top of a vertical wall. The point of projection is 8 m above horizontal ground. The ball hits a vertical fence which is at a horizontal distance of 9 m from the wall (see diagram).

1. Calculate the height above the ground of the point where the ball hits the fence.
2. Calculate the direction of motion of the ball immediately before it hits the fence.
3. It is given that \(30 \%\) of the kinetic energy of the ball is lost when it hits the fence. Calculate the speed of the ball immediately after it hits the fence.

1

1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
   1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
   2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
   3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

4

1. Two discs, P of mass 4 kg and Q of mass 5 kg , are sliding along the same line on a smooth horizontal plane when they collide. The velocity of P before the collision and the velocity of Q after the collision are shown in Fig. 4. P loses \(\frac { 5 } { 9 }\) of its kinetic energy in the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{71d839d8-12ca-4806-8f74-c572e7e21891-5_294_899_390_584} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Show that after the collision P has a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to its original motion. While colliding, the discs are in contact for \(\frac { 1 } { 5 } \mathrm {~s}\).
   2. Find the impulse on P in the collision and the average force acting on the discs.
   3. Find the velocity of Q before the collision and the coefficient of restitution between the two discs.
2. A particle is projected from a point 2.5 m above a smooth horizontal plane. Its initial velocity is \(5.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) below the horizontal, where \(\sin \theta = \frac { 15 } { 17 }\). The coefficient of restitution between the particle and the plane is \(\frac { 4 } { 5 }\).
   1. Show that, after bouncing off the plane, the greatest height reached by the particle is 2.5 m .
   2. Calculate the horizontal distance between the two points at which the particle is 2.5 m above the plane.

7. \begin{figure}[h] \end{figure} Figure 3 shows the path of a golf ball which is hit from the point \(O\) with speed \(49 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the ball is in a vertical plane containing \(O\) and the hole at which the ball is aimed. The hole is 170 m from \(O\) and on the same horizontal level as \(O\).

1. Suggest a suitable model for the motion of the golf ball. Find, correct to 3 significant figures,
2. the distance beyond the hole at which the ball hits the ground,
3. the magnitude and direction of the velocity of the ball when it is directly above the hole.

6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),

1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the time taken for \(T\) to reach \(X\).
3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
   (4 marks)

5 A ball is dropped from a height of 2.5 m above a horizontal floor. The ball bounces repeatedly on the floor.

1. Find the speed of the ball when it first hits the floor.
2. The coefficient of restitution between the ball and the floor is \(e\).
   1. Show that the time taken between the first contact of the ball with the floor and the second contact of the ball with the floor is \(\frac { 10 e } { 7 }\) seconds.
   2. Find, in terms of \(e\), the time taken between the second contact and the third contact of the ball with the floor.
3. Find, in terms of \(e\), the total vertical distance travelled by the ball from when it is dropped until its third contact with the floor.
4. State a modelling assumption for answering this question, other than the ball being a particle.

2 A particle of mass 0.3 kg is projected horizontally under gravity with velocity \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point 0.4 m above a smooth horizontal plane. The particle first hits the plane at point \(A\); it bounces and hits the plane a second time at point \(B\). The distance \(A B\) is 1 m . Calculate

1. the vertical component of the velocity of the particle when it arrives at \(A\), and the time taken for the particle to travel from \(A\) to \(B\),
2. the coefficient of restitution between the particle and the plane,
3. the impulse exerted by the plane on the particle at \(A\).

3. \begin{figure}[h] \end{figure} A particle of mass \(m\) is suspended at a point \(A\) vertically below a fixed point \(O\) by a light inextensible string of length \(a\) as shown in Figure 2. The particle is given a horizontal velocity \(u\) and subsequently moves along a circular arc until it reaches the point \(B\) where the string becomes slack. Given that the point \(B\) is at a height \(\frac { 1 } { 2 } a\) above the level of \(O\),

1. show that \(\angle B O A = 120 ^ { \circ }\),
2. show that \(u ^ { 2 } = \frac { 7 } { 2 } g a\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section \(A B\) followed by a downhill section \(B C\). The point \(O\) is on the same horizontal level as \(C\) and \(B C\) is a circular arc of radius 30 m and centre \(O\), such that \(\angle B O C = 90 ^ { \circ }\). A skier of mass 60 kg is skiing at \(12 \mathrm {~ms} ^ { - 1 }\) along \(A B\).

1. Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at \(B\).
   (4 marks)
   The skier subsequently leaves the slope at the point \(P\).
2. Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
3. Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point \(D\) which is on the same horizontal level as \(C\).

7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).

1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.

1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.

At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and sets a course in order to get as close to motorboat \(P\) as possible.

1. Find the course set by \(Q\), giving your answer as a bearing to the nearest degree.
2. Find the shortest distance between \(P\) and \(Q\).
3. Find the distance travelled by \(Q\) from its position at noon to the point of closest approach.
   \section*{June 2009}

At 12 noon, \(\operatorname { ship } A\) is 8 km due west of \(\operatorname { ship } B\). Ship \(A\) is moving due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Ship \(B\) is moving at a constant speed of \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing so that it passes as close to \(A\) as possible.

1. Find the bearing on which ship \(B\) moves.
2. Find the shortest distance between the two ships.
3. Find the time when the two ships are closest.
   

6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h] \end{figure}

1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
3. Find the value of \(\cos \theta\) correct to 3 significant figures.
4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.

4 A plane is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. A particle is projected from a point A on the plane with speed \(V \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(\phi ^ { \circ }\) with a line of greatest slope of the plane. The particle lands at a point B on the plane, as shown in the diagram, and the time of flight is \(T\) seconds. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-4_332_872_461_246} \begin{enumerate}[label=(\alph*)] \item By considering the motion of the particle perpendicular to the plane, show that \(\mathrm { T } = \frac { 2 \mathrm {~V} \sin \phi } { \mathrm {~g} \cos \theta }\). Consider the case when \(\theta = 30 , \phi = 25\) and \(V = 20\). \item

1. Calculate the distance AB .
2. State, with reasons but without any detailed calculations, what effect each of the following actions would have on the distance AB .

6 A section of a golf practice ground is inclined at \(15 ^ { \circ }\) to the horizontal. A golfer is hitting a ball up and down a line of greatest slope of this section of the practice ground. The golfer hits the ball up the slope, so that the ball initially makes an angle of \(30 ^ { \circ }\) with the slope. The ball first bounces on the slope 50 m from its point of projection.

1. Determine the initial speed of the ball. The golfer now hits the ball down the slope. The ball initially moves with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the ball initially travels at an angle \(\theta\) above the horizontal, as shown in Fig. 6. The ball first bounces at a point a distance \(L\) m down the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-6_545_791_794_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Show that \(\mathrm { L } = \frac { 800 } { \mathrm {~g} } \left( \frac { \sin \theta \cos \theta } { \cos 15 ^ { \circ } } + \frac { \sin 15 ^ { \circ } \cos ^ { 2 } \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\). You are given that \(\frac { \mathrm { dL } } { \mathrm { d } \theta } = \frac { 800 } { \mathrm {~g} } \left( \frac { \cos 2 \theta } { \cos 15 ^ { \circ } } - \frac { \sin 15 ^ { \circ } \sin 2 \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\).
3. Determine the value of \(\theta\) for which \(\frac { \mathrm { d } L } { \mathrm {~d} \theta } = 0\).
4. Hence determine the maximum distance the golfer can hit the ball down the slope.

7 A plane is inclined at \(30 ^ { \circ }\) above the horizontal. A particle is projected up the plane from a point C on the plane with a velocity of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above a line of greatest slope of the plane. The particle hits the plane at D. See Fig. 7. \begin{figure}[h] \end{figure}

1. Using the standard model for projectile motion, show that the time of flight, \(T\), is given by $$T = \frac { 28 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
2. Calculate the distance CD. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .

1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.

6. A ball is projected with velocity \(( 4 w \mathbf { i } + 7 w \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the top of a vertical tower. After 5 seconds, the ball hits the ground at a point that is 60 m horizontally from the foot of the tower. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.

1. Find the value of \(w\) and hence determine the height of the tower.
2. Determine the proportion of the 5 seconds for which the ball is on its way down.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).

1. Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
2. Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
3. Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).