3.02i Projectile motion: constant acceleration model

6 A particle is projected from a point \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) above the horizontal and it moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at any subsequent time, \(t\) seconds, are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$
   \includegraphics[max width=\textwidth, alt={}]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_551_575_667_826}
   The particle subsequently passes through the point \(A\) with coordinates \(( h , - h )\) as shown in the diagram. It is given that \(v = 14\) and \(\theta = 30 ^ { \circ }\).
2. Calculate \(h\).
3. Calculate the direction of motion of the particle at \(A\).
4. Calculate the speed of the particle at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_278_1061_1749_543} Two small spheres, \(P\) and \(Q\), are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface and the base of the cylinder. The mass of \(P\) is 0.2 kg , the mass of \(Q\) is 0.3 kg and the radius of the cylinder is \(0.4 \mathrm {~m} . P\) and \(Q\) are stationary at opposite ends of a diameter of the base of the cylinder (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(0.5 . P\) is given an impulse of magnitude 0.8 Ns in a tangential direction.

1. Calculate the speeds of the particles after \(P\) 's first impact with \(Q\). \(Q\) subsequently catches up with \(P\) and there is a second impact.
2. Calculate the speeds of the particles after this second impact.
3. Calculate the magnitude of the force exerted on \(Q\) by the curved surface of the cylinder after the second impact.

6 \includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle \(P\) is projected with speed \(V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 1 }\) from a point \(O\) on horizontal ground. When \(P\) is vertically above a point \(A\) on the ground its height is 250 m and its velocity components are \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards (see diagram).

1. Show that \(V _ { 1 } = 86.0\) and \(\theta _ { 1 } = 62.3 ^ { \circ }\), correct to 3 significant figures. At the instant when \(P\) is vertically above \(A\), a second particle \(Q\) is projected from \(O\) with speed \(V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 2 } . P\) and \(Q\) hit the ground at the same time and at the same place.
2. Calculate the total time of flight of \(P\) and the total time of flight of \(Q\).
3. Calculate the range of the particles and hence calculate \(V _ { 2 }\) and \(\theta _ { 2 }\).

6 A small ball \(B\) is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) from a point \(O\) on a horizontal plane, and moves freely under gravity.

1. Calculate the height of \(B\) above the plane when moving horizontally. \(B\) has mass 0.4 kg . At the instant when \(B\) is moving horizontally it receives an impulse of magnitude \(I \mathrm { Ns }\) in its direction of motion which immediately increases the speed of \(B\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate \(I\). For the instant when \(B\) returns to the plane, calculate
3. the speed and direction of motion of \(B\),
4. the time of flight, and the distance of \(B\) from \(O\).

1 A particle \(P\) is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of \(35 ^ { \circ }\) above the horizontal from a point \(O\). For the instant 3 s after projection, calculate the magnitude and direction of the velocity of \(P\).

7 A particle \(P\) is projected horizontally with speed \(15 \mathrm {~ms} ^ { - 1 }\) from the top of a vertical cliff. At the same instant a particle \(Q\) is projected from the bottom of the cliff, with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. \(P\) and \(Q\) move in the same vertical plane. The height of the cliff is 60 m and the ground at the bottom of the cliff is horizontal.

1. Given that the particles hit the ground simultaneously, find the value of \(\theta\) and find also the distance between the points of impact with the ground.
2. Given instead that the particles collide, find the value of \(\theta\), and determine whether \(Q\) is rising or falling immediately before this collision.

2 A particle is projected horizontally with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground.

8 A particle is projected with speed \(49 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
   \includegraphics[max width=\textwidth, alt={}]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_627_1249_1699_447}
   The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), and the corresponding points where the particle returns to the plane are \(A _ { 1 }\) and \(A _ { 2 }\) respectively (see diagram).
2. Find \(\theta _ { 1 }\) and \(\theta _ { 2 }\).
3. Calculate the distance between \(A _ { 1 }\) and \(A _ { 2 }\).

7 A small ball is projected at an angle of \(50 ^ { \circ }\) above the horizontal, from a point \(A\), which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.

1. Find the speed with which the ball is projected from \(A\). The ball hits a net at a point \(B\) when it has travelled a horizontal distance of 45 m .
2. Find the height of \(B\) above the ground.
3. Find the speed of the ball immediately before it hits the net.

1. At time \(t = 0\), a small ball is projected vertically upwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) that is 16.8 m above horizontal ground.

The speed of the ball at the instant immediately before it hits the ground for the first time is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The ball hits the ground for the first time at time \(t = T\) seconds.
The motion of the ball, from the instant it is projected until the instant just before it hits the ground for the first time, is modelled as that of a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Using the model,

1. show that \(U = 5\)
2. find the value of \(T\),
3. find the time from the instant the ball is projected until the instant when the ball is 1.2 m below \(A\).
4. Sketch a velocity-time graph for the motion of the ball for \(0 \leqslant t \leqslant T\), stating the coordinates of the start point and the end point of your graph. In a refinement of the model of the motion of the ball, the effect of air resistance on the ball is included and this refined model is now used to find the value of \(U\).
5. State, with a reason, how this new value of \(U\) would compare with the value found in part (a), using the initial unrefined model.
6. Suggest one further refinement that could be made to the model, apart from including air resistance, that would make the model more realistic.

1. The point \(A\) is 1.8 m vertically above horizontal ground.

At time \(t = 0\), a small stone is projected vertically upwards with speed \(U \mathrm {~ms} ^ { - 1 }\) from the point \(A\). At time \(t = T\) seconds, the stone hits the ground.
The speed of the stone as it hits the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) In an initial model of the motion of the stone as it moves from \(A\) to where it hits the ground

• the stone is modelled as a particle moving freely under gravity
• the acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\)

Using the model,

1. find the value of \(U\),
2. find the value of \(T\).
3. Suggest one refinement, apart from including air resistance, that would make the model more realistic. In reality the stone will not move freely under gravity and will be subject to air resistance.
4. Explain how this would affect your answer to part (a).

1. A small stone is projected vertically upwards with speed \(39.2 \mathrm {~ms} ^ { - 1 }\) from a point \(O\).

The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10s later. Using the model, find

1. the height of \(O\) above the ground,
2. the total length of time for which the speed of the stone is less than or equal to \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. State one refinement that could be made to the model that would make your answer to part (a) more accurate.

5. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) lie 50 m apart on horizontal ground.
At time \(t = 0\) two small balls, \(P\) and \(Q\), are projected in the vertical plane containing \(A B\).
Ball \(P\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(30 ^ { \circ }\) to \(A B\).
Ball \(Q\) is projected from \(B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 3.
At time \(t = 2\) seconds, \(P\) and \(Q\) collide.
Until they collide, the balls are modelled as particles moving freely under gravity.

1. Find the velocity of \(P\) at the instant before it collides with \(Q\).
2. Find
   1. the size of angle \(\theta\),
   2. the value of \(u\).
3. State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.

5. \begin{figure}[h] \end{figure} A golf ball is at rest at the point \(A\) on horizontal ground.
The ball is hit and initially moves at an angle \(\alpha\) to the ground.
The ball first hits the ground at the point \(B\), where \(A B = 120 \mathrm {~m}\), as shown in Figure 3.
The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using this model,

1. show that \(U ^ { 2 } \sin \alpha \cos \alpha = 588\) The ball reaches a maximum height of 10 m above the ground.
2. Show that \(U ^ { 2 } = 1960\) In a refinement to the model, the effect of air resistance is included.
   The motion of the ball, from \(A\) to \(B\), is now modelled as that of a particle whose initial speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This refined model is used to calculate a value for \(V\)
3. State which is greater, \(U\) or \(V\), giving a reason for your answer.
4. State one further refinement to the model that would make the model more realistic.

5. \begin{figure}[h] \end{figure} A small ball is projected with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on horizontal ground. After moving for \(T\) seconds, the ball passes through the point \(A\). The point \(A\) is 40 m horizontally and 20 m vertically from the point \(O\), as shown in Figure 2. The motion of the ball from \(O\) to \(A\) is modelled as that of a particle moving freely under gravity. Given that the ball is projected at an angle \(\alpha\) to the ground, use the model to

1. show that \(T = \frac { 10 } { 7 \cos \alpha }\)
2. show that \(\tan ^ { 2 } \alpha - 4 \tan \alpha + 3 = 0\)
3. find the greatest possible height, in metres, of the ball above the ground as the ball moves from \(O\) to \(A\). The model does not include air resistance.
4. State one other limitation of the model.

5. \begin{figure}[h] \end{figure} At time \(t = 0\), a small stone is projected with velocity \(35 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The stone is projected at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) In an initial model

• the stone is modelled as a particle \(P\) moving freely under gravity
• the stone hits the ground at the point \(A\)

Figure 4 shows the path of \(P\) from \(O\) to \(A\).
For the motion of \(P\) from \(O\) to \(A\)

• at time \(t\) seconds, the horizontal distance of \(P\) from \(O\) is \(x\) metres
• at time \(t\) seconds, the vertical distance of \(P\) above the ground is \(y\) metres
  1. Using the model, show that

$$y = \frac { 3 } { 4 } x - \frac { 1 } { 160 } x ^ { 2 }$$

Use the answer to (a), or otherwise, to find the length \(O A\). Using the model, the greatest height of the stone above the ground is found to be \(H\) metres.

Use the answer to (a), or otherwise, to find the value of \(H\).

Using this new model, the greatest height of the stone above the ground is found to be \(K\) metres.

State which is greater, \(H\) or \(K\), justifying your answer.

State one limitation of this refined model.

5. \begin{figure}[h] \end{figure} A small ball is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The point \(O\) is 25 m vertically above the point \(N\) which is on horizontal ground. The ball is projected at an angle of \(45 ^ { \circ }\) above the horizontal.
The ball hits the ground at a point \(A\), where \(A N = 100 \mathrm {~m}\), as shown in Figure 2 .
The motion of the ball is modelled as that of a particle moving freely under gravity.
Using this initial model,

1. show that \(U = 28\)
2. find the greatest height of the ball above the horizontal ground \(N A\). In a refinement to the model of the motion of the ball from \(O\) to \(A\), the effect of air resistance is included. This refined model is used to find a new value of \(U\).
3. How would this new value of \(U\) compare with 28, the value given in part (a)?
4. State one further refinement to the model that would make the model more realistic. \section*{" " \(_ { \text {" } } ^ { \text {" } }\) " "}

4. \begin{figure}[h] \end{figure} A small stone is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
Point \(O\) is 70 m vertically above the point \(N\).
Point \(N\) is on horizontal ground.
The stone is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The stone hits the ground at the point \(A\), as shown in Figure 3.
The stone is modelled as a particle moving freely under gravity.
The acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 m ~ s } \mathbf { m ~ } ^ { \mathbf { - 2 } }\) Using the model,

1. find the time taken for the stone to travel from \(O\) to \(A\),
2. find the speed of the stone at the instant just before it hits the ground at \(A\). One limitation of the model is that it ignores air resistance.
3. State one other limitation of the model that could affect the reliability of your answers.

9 A pebble is thrown horizontally at \(14 \mathrm {~ms} ^ { - 1 }\) from a window which is 5 m above horizontal ground. The pebble goes over a fence 2 m high \(d \mathrm {~m}\) away from the window as shown in Fig. 9. The origin is on the ground directly below the window with the \(x\)-axis horizontal in the direction in which the pebble is thrown and the \(y\)-axis vertically upwards. \begin{figure}[h] \end{figure}

1. Find the time the pebble takes to reach the ground.
2. Find the cartesian equation of the trajectory of the pebble.
3. Find the range of possible values for \(d\).

9 A cannonball is fired from a point on horizontal ground at \(100 \mathrm {~ms} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate

1. the greatest height the cannonball reaches,
2. the range of the cannonball.

7 In this question the \(x\) - and \(y\)-directions are horizontal and vertically upwards respectively and the origin is on horizontal ground.
A ball is thrown from a point 5 m above the origin with an initial velocity \(\binom { 14 } { 7 } \mathrm {~ms} ^ { - 1 }\).

1. Find the position vector of the ball at time \(t \mathrm {~s}\) after it is thrown.
2. Find the distance between the origin and the point at which the ball lands on the ground.

15 A projectile is launched from a point on level ground with an initial velocity \(u\) at an angle \(\theta\) above the horizontal.

1. Show that the range of the projectile is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos \theta } { g }\).
2. Determine the set of values of \(\theta\) for which the maximum height of the projectile is greater than the range, where \(\theta\) is an acute angle. Give your answer in degrees.

14 A man runs at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. A woman is standing on a bridge that spans the road. At the instant that the man passes directly below the woman she throws a ball with initial speed \(u \mathrm {~ms} ^ { - 1 }\) at \(\alpha ^ { \circ }\) above the horizontal. The path of the ball is directly above the road. The man catches the ball 2.4 s after it is thrown. At the instant the man catches it, the ball is 3.6 m below the level of the point of projection.

1. Explain what it means that the ball is modelled as a particle.
2. Find the vertical component of the ball's initial velocity.
3. Find each of the following.

13 A projectile is fired from ground level at \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal.

1. State a modelling assumption that is used in the standard projectile model.
2. Find the cartesian equation of the trajectory of the projectile. The projectile travels above horizontal ground towards a wall that is 110 m away from the point of projection and 5 m high. The projectile reaches a maximum height of 22.5 m .
3. Determine whether the projectile hits the wall.

7 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A small stone is projected from a point O with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7.
You are given that \(\sin \theta = \frac { 12 } { 13 }\).
After \(t\) seconds the horizontal displacement of the stone from O is \(x\) metres and the vertical displacement is \(y\) metres. \begin{figure}[h] \end{figure}

1. Using the standard model for projectile motion,
   The stone passes through a point A . Point A is 16 m above the level of O .
2. Find the two possible horizontal distances of A from O . A toy balloon is projected from O with the same initial velocity as the small stone.
3. Suggest two ways in which the standard model could be adapted.

7 \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\). \(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).

1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}