3.02i Projectile motion: constant acceleration model

7. \begin{figure}[h] \end{figure} A shot is projected upwards from the top of a cliff with a velocity of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. It strikes the ground 52.5 m vertically below the level of the point of projection, as shown in Fig. 3. The motion of the shot is modelled as that of a particle moving freely under gravity. Find, to 3 significant figures,

1. the horizontal distance from the point of projection at which the shot strikes the ground,
2. the speed of the shot as it strikes the ground.

6. A cricket ball is hit from a height of 0.8 m above horizontal ground with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The motion of the ball is modelled as that of a particle moving freely under gravity.

1. Find, to 2 significant figures, the greatest height above the ground reached by the ball. When the ball has travelled a horizontal distance of 36 m , it hits a window.
2. Find, to 2 significant figures, the height above the ground at which the ball hits the window.
3. State one physical factor which could be taken into account in any refinement of the model which would make it more realistic. Figure 2

7. \begin{figure}[h] \end{figure} A hollow sphere has internal radius \(r\) and centre \(O\). A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point \(B\) is the lowest point of the inner surface of the bowl. The point \(A\), where angle \(A O B = 120 ^ { \circ }\), lies on the rim of the bowl, as shown in Figure 4. A particle \(P\) of mass \(m\) is projected from \(A\), with speed \(U\) at \(90 ^ { \circ }\) to \(O A\), and moves on the smooth inner surface of the bowl. The motion of \(P\) takes place in the vertical plane \(O A B\).

1. Find, in terms of \(m , g , U\) and \(r\), the magnitude of the force exerted on \(P\) by the bowl at the instant when \(P\) passes through \(B\).
2. Find, in terms of \(g , U\) and \(r\), the greatest height above the floor reached by \(P\). Given that \(U > \sqrt { 2 g r }\)
3. show that, after leaving the surface of the bowl, \(P\) does not fall back into the bowl.

6. \begin{figure}[h] \end{figure} Figure 5 shows a hollow sphere, with centre \(O\) and internal radius \(a\), which is fixed to a horizontal surface. A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { \frac { 7 a g } { 2 } }\) from the lowest point \(A\) of the inner surface of the sphere. The particle moves in a vertical circle with centre \(O\) on the smooth inner surface of the sphere. The particle passes through the point \(B\), on the inner surface of the sphere, where \(O B\) is horizontal.

1. Find, in terms of \(m\) and \(g\), the normal reaction exerted on \(P\) by the surface of the sphere when \(P\) is at \(B\). The particle leaves the inner surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta , \theta > 0\), with the upward vertical.
2. Show that, after leaving the surface of the sphere at \(C\), the particle is next in contact with the surface at \(A\).

   END

7. \begin{figure}[h] \end{figure} A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).

1. Show that \(e \geqslant \frac { 1 } { 2 }\) The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\) As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
   Given that \(e = \frac { \sqrt { 3 } } { 2 }\)
2. find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).

7. \begin{figure}[h] \end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.

1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).

6. \begin{figure}[h] \end{figure} A smooth hollow cylinder of internal radius \(a\) is fixed with its axis horizontal. A particle \(P\) moves on the inner surface of the cylinder in a vertical circle with radius \(a\) and centre \(O\), where \(O\) lies on the axis of the cylinder. The particle is projected vertically downwards with speed \(u\) from point \(A\) on the circle, where \(O A\) is horizontal. The particle first loses contact with the cylinder at the point \(B\), where \(\angle A O B = 150 ^ { \circ }\), as shown in Figure 3. Given that air resistance can be ignored,

1. show that the speed of \(P\) at \(B\) is \(\sqrt { } \left( \frac { a g } { 2 } \right)\),
2. find \(u\) in terms of \(a\) and \(g\). After losing contact with the cylinder, \(P\) crosses the diameter through \(A\) at the point \(D\). At \(D\) the velocity of \(P\) makes an angle \(\theta ^ { \circ }\) with the horizontal.
3. Find the value of \(\theta\).

5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
2. the value of \(\theta\) when \(P\) loses contact with the sphere,
3. the speed of \(P\) as it hits the table.
   (Total 13 marks)

5. \begin{figure}[h] \end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(r\), forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at \(A\) with the rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\), where \(\angle A O B = \alpha\) and \(\tan \alpha = \frac { 3 } { 4 }\), is on the rim of the bowl, as shown in Figure 2. A small smooth marble \(M\) is placed inside the bowl at \(A\), and given an initial horizontal speed \(\sqrt { } ( g r )\). The motion of \(M\) takes place in the vertical plane \(O A B\).

1. Show that the speed of \(M\) as it reaches \(B\) is \(\sqrt { } \left( \frac { 3 } { 5 } g r \right)\). After leaving the surface of the bowl at \(B , M\) moves freely under gravity and first strikes the horizontal surface at the point \(C\). Given that \(r = 0.4 \mathrm {~m}\),
2. find the distance \(A C\).

5 A ball is thrown towards a hedge. Its position relative to the point from which it was thrown is given by the parametric equations $$x = 8 t , y = 10 t - 5 t ^ { 2 }$$

1. Find the cartesian equation of the trajectory of the ball.
2. The ball just clears the hedge. What can you say about the height of the hedge?

5.(a)The box below shows a student's attempt to prove the following identity for \(a > b > 0\) $$\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }$$ Let \(x = \arctan a\) and \(y = \arctan b\) ,so that \(a = \tan x\) and \(b = \tan y\) $$\begin{aligned} \text { So } \tan ( \arctan a - \arctan b ) & \equiv \tan ( x - y ) \\ & \equiv \frac { \tan x - \tan y } { 1 - \tan ^ { 2 } ( x y ) } \\ & \equiv \frac { a - b } { 1 - ( a b ) ^ { 2 } } \\ & \equiv \frac { a - a b + a b - b } { ( 1 - a b ) ( 1 + a b ) } \\ & \equiv \frac { a ( 1 - a b ) - b ( 1 - a b ) } { ( 1 - a b ) ( 1 + a b ) } \\ & \equiv \frac { a - b } { 1 + a b } \end{aligned}$$ Taking arctan of both sides gives \(\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }\) as required. There are three errors in the proof where the working does not follow from the previous line.

1. Describe these three errors.
2. Write out a correct proof of the identity.
   (b)[In this question take \(g\) to be \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4d5b914c-28b2-4485-a42e-627c95fa16e2-22_244_1267_1870_504} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Balls are projected,one after another,from a point,\(A\) ,one metre above horizontal ground. Each ball travels in a vertical plane towards a 6 metre high vertical wall of negligible thickness,which is a horizontal distance of \(10 \sqrt { 2 }\) metres from \(A\) . The balls are modelled as particles and it is assumed that there is no air resistance.
   Each ball is projected with an initial speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at a random angle \(\theta\) to the horizontal,where \(0 < \theta < 90 ^ { \circ }\) Given that a ball will pass over the wall precisely when \(\alpha \leqslant \theta \leqslant \beta\)
3. find, in degrees, the angle \(\beta - \alpha\)
4. Deduce that the probability that a particular ball will pass over the wall is \(\frac { 2 } { 3 }\)
5. Hence find the probability that exactly 2 of the first 10 balls projected pass over the wall. You should give your answer in the form \(\frac { P } { Q ^ { k } }\) where \(P , Q\) and \(k\) are integers and \(P\) is not a multiple of \(Q\).
6. Explain whether taking air resistance into account would increase or decrease the probability in (b)(iii).
7. find, in degrees, the angle \(\beta - \alpha\)

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find

1. the value of \(\theta\),
2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
3. Find the value of \(T\).

4. A darts player throws darts at a dart board which hangs vertically. The motion of a dart is modelled as that of a particle moving freely under gravity. The darts move in a vertical plane which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits the board at a point which is 10 cm below the level from which it was thrown.

1. Find the horizontal distance from the point where the dart was thrown to the dart board. The darts player moves his position. He now throws a dart from a point which is at a horizontal distance of 2.5 m from the board. He throws the dart at an angle of elevation \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). This dart hits the board at a point which is at the same level as the point from which it was thrown.
2. Find the speed with which the dart is thrown.

4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \(\left( 9.8 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) higher than O . When descending, the stone hits a plum which is 3.675 m higher than O . Air resistance should be neglected. Calculate the horizontal distance of the plum from O .

6 A small ball is kicked off the edge of a jetty over a calm sea. Air resistance is negligible. Fig. 6 shows

• the point of projection, O,
• the initial horizontal and vertical components of velocity,
• the point A on the jetty vertically below O and at sea level,
• the height, OA, of the jetty above the sea.

\begin{figure}[h] \end{figure} The time elapsed after the ball is kicked is \(t\) seconds.

1. Find an expression in terms of \(t\) for the height of the ball above O at time \(t\). Find also an expression for the horizontal distance of the ball from O at this time.
2. Determine how far the ball lands from A .

5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(35 ^ { \circ }\) to the horizontal. Assuming that air resistance on the object may be neglected and that the speed of sound in air is \(343 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate how long after projection the noise is heard at O .

6 A football is kicked with speed \(31 \mathrm {~ms} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) to the horizontal. It travels towards the goal which is 50 m away. The height of the crossbar of the goal is 2.44 m .

1. Does the ball go over the top of the crossbar? Justify your answer.
2. State one assumption that you made in answering part (i).

2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h] \end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.

1. Find its initial speed.
2. Find the ball's flight time and range, \(R \mathrm {~m}\).
3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
   (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.

1. Find the value of \(\lambda\).
2. Find
   1. the speed of \(P\) at the instant when it passes through \(A\),
   2. the direction of motion of \(P\) at the instant when it passes through \(A\).
      HMAV SIHI NITIIIUM ION OC
      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC

7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h] \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)

8 A missile is projected with initial speed \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate

1. the maximum height of the missile above the level of the point of projection,
2. the distance of the missile from the point of projection at the instant when it is moving downwards at an angle of \(10 ^ { \circ }\) to the horizontal.

1 A ball is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(55 ^ { \circ }\) above the horizontal. At the instant when the ball reaches its greatest height, it hits a vertical wall, which is perpendicular to the ball's path. The coefficient of restitution between the ball and the wall is 0.65 . Calculate the speed of the ball

1. immediately before its impact with the wall,
2. immediately after its impact with the wall.

7 A missile is projected from a point \(O\) on horizontal ground with speed \(175 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The horizontal lower surface of a cloud is 650 m above the ground.

1. Find the value of \(\theta\) for which the missile just reaches the cloud. It is given that \(\theta = 55 ^ { \circ }\).
2. Find the length of time for which the missile is above the lower surface of the cloud.
3. Find the speed of the missile at the instant it enters the cloud.

1 A stone is projected from a point on level ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(\theta ^ { \circ }\) above the horizontal. When the stone is at its greatest height it just passes over the top of a tree that is 17 m high. Calculate \(\theta\).