3.02i Projectile motion: constant acceleration model

7 A particle \(P\) is projected with speed \(\mathrm { Vms } ^ { - 1 }\) at an angle \(75 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane. It then moves freely under gravity.

1. Show that the total time of flight, in seconds, is \(\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }\).
   A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from \(O\). The particle is projected as before but now strikes the barrier, rebounds and returns to \(O\). The coefficient of restitution between the barrier and the particle is \(\frac { 3 } { 5 }\).
2. Explain why the total time of flight is unchanged.
3. Find an expression for \(V\) in terms of \(g\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A 10 \mathrm {~m}\) above horizontal ground. The angle of projection is \(55 ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 3. Find

1. the speed of \(P\) as it hits the ground at \(B\),
2. the direction of motion of \(P\) as it hits the ground at \(B\),
3. the time taken for \(P\) to move from \(A\) to \(B\).

7. A particle is projected from a point \(O\) with speed \(U\) at an angle of elevation \(\alpha\) to the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).

1. Show that $$y = x \tan \alpha - \frac { g x ^ { 2 } \left( 1 + \tan ^ { 2 } \alpha \right) } { 2 U ^ { 2 } }$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-22_330_857_632_548} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A small stone is projected horizontally with speed \(U\) from a point \(C\) at the top of a vertical cliff \(A C\) so as to hit a fixed target \(B\) on the horizontal ground. The point \(C\) is a height \(h\) above the ground, as shown in Figure 3. The time of flight of the stone from \(C\) to \(B\) is \(T\), and the stone is modelled as a particle moving freely under gravity.
2. Find, in terms of \(U , g\) and \(T\), the speed of the stone as it hits the target at \(B\). It is found that, using the same initial speed \(U\), the target can also be hit by projecting the stone from \(C\) at an angle \(\alpha\) above the horizontal. The stone is again modelled as a particle moving freely under gravity and the distance \(A B = d\).
3. Using the result in part (a), or otherwise, show that $$d = \frac { 1 } { 2 } g T ^ { 2 } \tan \alpha$$

7. \begin{figure}[h] \end{figure} At time \(t = 0\) a particle \(P\) is projected from a fixed point \(A\) on horizontal ground. The particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the ground. The particle moves freely under gravity. At time \(t = 3\) seconds, \(P\) is passing through the point \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at an angle \(\beta\) to the horizontal, as shown in Figure 5.

1. By considering energy, find the height of \(B\) above the ground.
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the least speed of \(P\) as \(P\) travels from \(A\) to \(B\). As \(P\) travels from \(A\) to \(B\), the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 15\) for an interval of \(T\) seconds.
5. Find the value of \(T\).
   \section*{\textbackslash section*\{Question 7 continued\}}

7. \begin{figure}[h] \end{figure} The fixed point \(A\) is 20 m vertically above the point \(O\) which is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The particle moves freely under gravity. At time \(t = 5\) seconds, \(P\) strikes the ground at the point \(B\), where \(O B = 40 \mathrm {~m}\), as shown in Figure 4.

1. By considering energy, find the speed of \(P\) as it hits the ground at \(B\).
2. Find the least speed of \(P\) as it moves from \(A\) to \(B\).
3. Find the length of time for which the speed of \(P\) is more than \(10 \mathrm {~ms} ^ { - 1 }\).

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 girl throws a small stone with initial speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h] \end{figure}

1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
2. (A) Find the time it takes for the stone to reach its highest point.
   (B) Calculate the distance of O from the base of the wall.
   (C) Show that the stone passes over the wall with 2.5 m clearance.
3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
4. Calculate \(d\).

4 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$ where distances are in metres and the origin is a point on the level ground.

1. Write down
   (A) the height from which P is projected,
   (B) the value of \(g\) in this model.
2. Find the displacement of P from \(t = 3\) to \(t = 5\).
3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$

5 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Find the time it takes for the stone to reach the water.
2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.

5 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .

1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of projection is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the angle of projection of the stone.
3. Find the horizontal range of the stone. Section B (36 marks)

8 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).

1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
2. Calculate the maximum height reached by the ball.
3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
4. Determine the following when \(t = 1.25\).
   (A) The vertical component of the velocity of the ball.
   (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
   (C) The speed of the ball.
5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.

6 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A golf ball is hit from ground level over horizontal ground. The initial velocity of the ball is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). Air resistance may be neglected.

1. Find an expression for the height of the ball above the ground \(t\) seconds after projection.
2. Calculate the horizontal range of the ball. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_358_447_360_849} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
   \end{figure} A box of mass 8 kg is supported by a continuous light string ACB that is fixed at A and at B and passes through a smooth ring on the box at C, as shown in Fig. 7.1. The box is in equilibrium and the tension in the string section AC is 60 N .

4 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
2. Show that \(\cos \alpha = 0.28\).
3. Calculate the greatest height reached by the particle.

4 Fig. 4 illustrates a situation in which a film is being made. A cannon is fired from the top of a vertical cliff towards a ship out at sea. The director wants the cannon ball to fall just short of the ship so that it appears to be a near-miss. There are actors on the ship so it is important that it is not hit by mistake. The cannon ball is fired from a height 75 m above the sea with an initial velocity of \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. The ship is 90 m from the bottom of the cliff. \begin{figure}[h] \end{figure}

1. The director calculates where the cannon ball will hit the sea, using the standard projectile model and taking the value of \(g\) to be \(10 \mathrm {~ms} ^ { - 2 }\). Verify that according to this model the cannon ball is in the air for 5 seconds. Show that it hits the water less than 5 m from the ship.
2. Without doing any further calculations state, with a brief reason, whether the cannon ball would be predicted to travel further from the cliff if the value of \(g\) were taken to be \(9.8 \mathrm {~ms} ^ { - 2 }\).

5 A golf ball is hit at an angle of \(60 ^ { \circ }\) to the horizontal from a point, O , on level horizontal ground. Its initial speed is \(20 \mathrm {~ms} ^ { - 1 }\). The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time \(t \mathrm {~s}\) the horizontal and vertical components of its displacement from O are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\).

1. Write down equations for \(x\) and \(y\) in terms of \(t\).
2. Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 } .$$
3. Find the range of the golf ball.
4. A bird is hovering at position \(( 20,16 )\). Find whether the golf ball passes above it, passes below it or hits it.

5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5. \begin{figure}[h] \end{figure}

• The stone is thrown from point O on level ground. Its initial velocity is \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(8 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.
• The pigeon is at point P which is 4 m above the ground.
• The house is 22.5 m from O .
• The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.

Show that the stone does not reach the height of the pigeon. Determine whether the stone hits the window.

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

7. A particle \(P\) is projected from a fixed point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. When \(P\) passes through the point \(B\) on its path, it has speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the vertical distance between \(A\) and \(B\). The minimum speed of \(P\) on its path from \(A\) to \(B\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the size of angle \(\alpha\).
3. Find the horizontal distance between \(A\) and \(B\).

6. [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

6. A particle \(P\) is projected from a fixed point \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. As \(P\) passes through the point \(B\) on its path, \(P\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\beta\) below the horizontal.

1. By considering energy, find the vertical distance between \(A\) and \(B\). Particle \(P\) takes 1.5 seconds to travel from \(A\) to \(B\).
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the length of time for which the speed of \(P\) is less than \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h] \end{figure} At time \(t = 0\), a small ball is projected from a fixed point \(O\) on horizontal ground. The ball is projected from \(O\) with velocity ( \(p \mathbf { i } + q \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(p\) and \(q\) are positive constants. The ball moves freely under gravity. At time \(t = 3\) seconds, the ball passes through the point \(A\) with velocity ( \(8 \mathbf { i } - 12 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 4.

1. Find the speed of the ball at the instant it is projected from \(O\). For an interval of \(T\) seconds the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the ball is such that \(v \leqslant 10\)
2. Find the value of \(T\). At the point \(B\) on the path of the ball, the direction of motion of the ball is perpendicular to the direction of motion of the ball at \(A\).
3. Find the vertical height of \(B\) above \(A\).

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

\begin{figure}[h] \end{figure} A small ball is projected with velocity \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(A\) on horizontal ground. The ball hits the ground at the point \(B\), as shown in Figure 5. The motion of the ball is modelled as a particle moving freely under gravity.

1. Find the distance \(A B\). When the height of the ball above the ground is more than \(h\) metres, the speed of the ball is less than \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the smallest possible value of \(h\). When the ball is at the point \(C\) on its path, the direction of motion of the ball is perpendicular to the direction of motion of the ball at the instant before it hits the ground at \(B\).
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of the ball when it is at \(C\).