Deriving trajectory equation

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.

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 small ball is thrown horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the roof of a building. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically downwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence show that the equation of the trajectory of the ball is \(y = 0.2 x ^ { 2 }\). The ball strikes the horizontal ground which surrounds the building at a point \(A\).
2. Given that \(O A = 18 \mathrm {~m}\), calculate the value of \(x\) at \(A\), and the speed of the ball immediately before it strikes the ground at \(A\).

3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = 2 x - x ^ { 2 }$$

1. Find the value of \(x\) for the point where \(P\) strikes the plane.
2. Find the angle and speed of projection of \(P\).
3. Calculate the speed of \(P\) immediately before it strikes the plane.

4 A small ball is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) 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 \(t\) and hence find the equation of the trajectory of the ball.
2. Find \(x\) for the position of the ball when its path makes an angle of \(15 ^ { \circ }\) below the horizontal. [4]

1 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the vertically upwards displacement of the ball from \(O\) is \(\left( 14 t - k t ^ { 2 } \right) \mathrm { m }\), where \(k\) is a constant.

1. State the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-03_56_1563_495_331}
2. Show that the initial speed of the ball is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the horizontal displacement of the ball from \(O\) when \(t = 3\).

3 A small ball is projected 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, where \(x = 4 t\) and \(y = 6 t - 5 t ^ { 2 }\).

1. Find the equation of the trajectory of the ball.
2. Hence or otherwise calculate the angle of projection of the ball and its initial speed.

3 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of \(P\) is \(y = x - 0.016 x ^ { 2 }\).
2. Calculate the horizontal distance between the two positions at which \(P\) is 2.4 m above the ground.

4 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the trajectory of \(P\) is $$y = \frac { x } { \sqrt { 3 } } - \frac { 4 x ^ { 2 } } { 375 }$$
2. Find the horizontal distance between the two points at which \(P\) is 5 m above the ground.

4 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
2. Find the value of \(x\) for which \(O B\) makes an angle of \(45 ^ { \circ }\) above the horizontal.

2 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. 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. It is given that \(x = 40 t\).

1. Calculate the initial speed of the ball, and express \(y\) in terms of \(t\).
2. Hence find the equation of the trajectory of the ball.

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.

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.

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.
5. 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.
6. Calculate the speeds of the particles after this second impact.
7. Calculate the magnitude of the force exerted on \(Q\) by the curved surface of the cylinder after the second impact.

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.

1 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 \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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.

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.

5 A particle is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\) from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the particle from \(O\) at any subsequent time \(t \mathrm {~s}\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the particle.
2. Calculate the values of \(x\) when \(y = 0.6\).
3. Find the direction of motion of the particle when \(y = 0.6\) and the particle is rising.

5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.

1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
   (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
3. State two modelling assumptions that you have made.

5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.

1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
   (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-3_330_1411_1902_303}
3. State two modelling assumptions that you have made.

2 A projectile is launched from a point \(O\) on top of a cliff with initial velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\) and moves in a vertical plane. During the motion, the position vector of the projectile relative to the point \(O\) is \(( x \mathbf { i } + y \mathbf { j } )\) metres where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively.

1. Show that, during the motion, the equation of the trajectory of the projectile is given by $$y = x \tan \alpha - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$$
2. When \(u = 21\) and \(\alpha = 55 ^ { \circ }\), the projectile hits a small buoy \(B\). The buoy is at a distance \(s\) metres vertically below \(O\) and at a distance \(s\) metres horizontally from \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-04_601_935_964_548}
   1. Find the value of \(s\).
   2. Find the acute angle between the velocity of the projectile and the horizontal just before the projectile hits \(B\), giving your answer to the nearest degree.
      [0pt] [5 marks]

10 A small ball \(P\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the ball from \(O\) at any subsequent time \(t\) seconds are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The ball is modelled as a particle and the acceleration due to gravity is taken to be \(10 \mathrm {~ms} ^ { - 2 }\).

1. Show that the equation of the trajectory of \(P\) is $$y = x \tan \theta - \frac { x ^ { 2 } } { 5 } \left( 1 + \tan ^ { 2 } \theta \right)$$ It is given that \(\tan \theta = 3\).
2. Using part (i), find the maximum height above the level of \(O\) of \(P\) in the subsequent motion.
3. Find the values of \(t\) when \(P\) is moving at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 1 } { 3 }\).
4. Give two possible reasons why the values of \(t\) found in part (iii) may not be accurate. \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-09_435_714_267_678} Two particles \(P\) and \(Q\), with masses 2 kg and 8 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. Plane \(\Pi _ { 1 }\) is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and plane \(\Pi _ { 2 }\) is inclined at an angle of \(\theta\) to the horizontal. Particle \(P\) is on \(\Pi _ { 1 }\) and \(Q\) is on \(\Pi _ { 2 }\) with the string taut (see diagram). \(\Pi _ { 1 }\) is rough and the coefficient of friction between \(P\) and \(\Pi _ { 1 }\) is \(\frac { \sqrt { 3 } } { 3 }\). \(\Pi _ { 2 }\) is smooth.
   The particles are released from rest and \(P\) begins to move towards the pulley with an acceleration of \(g \cos \theta \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
5. Show that \(\theta\) satisfies the equation $$4 \sin \theta - 5 \cos \theta = 1 .$$
6. By expressing \(4 \sin \theta - 5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), find, correct to 3 significant figures, the tension in the string.

2 A particle is projected from a point \(O\) on a horizontal plane and has initial velocity components of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) parallel to and perpendicular to the plane respectively. At time \(t\) seconds after projection, the horizontal and upward vertical distances of the particle from the point \(O\) are \(x\) metres and \(y\) metres respectively.

1. Show that \(x\) and \(y\) satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
2. By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
3. Hence find the time for which the particle is more than 1 metre above the plane.

8 A particle is projected from a point \(O\) with initial speed \(U\) at an angle \(\theta\) above the horizontal. At time \(t\) after projection the position of the particle is \(( x , y )\) relative to horizontal and vertical axes through \(O\).

1. Write down expressions for \(x\) and \(y\) at time \(t\). Hence derive the cartesian equation of the trajectory of the particle.
2. A player in a cricket match throws the ball with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to another player who is 45 metres away. Assume that the players throw and catch the ball at the same height above the ground. Show that there are two possible trajectories and find their respective angles of projection. [4]
3. Describe briefly one advantage of each trajectory.

14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).

1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$