Horizontal projection from height

1 A particle is projected horizontally with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a high cliff. Find the direction of motion of the particle after 2 s .

2 A stone is thrown with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from the top of a vertical cliff 20 m above the sea. Calculate

1. the distance from the foot of the cliff to the point where the stone enters the sea,
2. the speed of the stone when it enters the sea.

3 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_502_789_1742_680} A stone is projected horizontally, with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from the top of a vertical cliff of height 45 m above sea level (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of the stone from the top of the cliff are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the stone's trajectory.
2. Find the angle the trajectory makes with the horizontal at the point where the stone reaches sea level.

1 A stone is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The horizontal and vertically upward displacements of the stone from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Find the equation of the stone's trajectory. The stone enters the sea at a horizontal distance of 24 m from the base of the cliff.
2. Find the height above sea level of the top of the cliff.

1 A stone \(S\) is thrown horizontally from the top \(T\) of a high tower. At the instant 1.6 s after \(S\) is thrown, the line \(S T\) makes an angle of \(30 ^ { \circ }\) below the horizontal. Find the speed with which \(S\) is thrown. [3]

4 A small object is projected horizontally with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object 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 the object is \(y = - \frac { 5 x ^ { 2 } } { V ^ { 2 } }\).
   The object passes through points with coordinates \(( a , - a )\) and \(\left( a ^ { 2 } , - 16 a \right)\), where \(a\) is a positive constant.
2. Find the value of \(a\).
3. Given that the object strikes the ground at the point where \(x = 5 a\), find the height of \(O\) above the ground .

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 $$d = \frac { 1 } { 2 } g T ^ { 2 } \tan \alpha$$

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.

7. \begin{figure}[h] \end{figure} A ball is thrown from a point \(A\) at a target, which is on horizontal ground. The point \(A\) is 12 m above the point \(O\) on the ground. The ball is thrown from \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal. The ball is modelled as a particle and the target as a point \(T\). The distance \(O T\) is 15 m . The ball misses the target and hits the ground at the point \(B\), where \(O T B\) is a straight line, as shown in Figure 4. Find

1. the time taken by the ball to travel from \(A\) to \(B\),
2. the distance \(T B\). The point \(X\) is on the path of the ball vertically above \(T\).
3. Find the speed of the ball at \(X\).

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.

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 .

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 girl stands at the edge of a quay and sees a tin can floating in the water. The water level is 5 metres below the top of the quay and the can is at a horizontal distance of 10 metres from the quay, as shown in the diagram.
The girl decides to throw a stone at the can. She throws the stone from a height of 1 metre above the top of the quay. The initial velocity of the stone is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) below the horizontal, so that the initial velocity of the stone is directed at the can, as shown in the diagram.
Assume that the stone is a particle and that it experiences no air resistance as it moves.

1. Find \(\alpha\).
2. Find the time that it takes for the stone to reach the level of the water.
3. Find the distance between the stone and the can when the stone hits the water.

6 A bullet is fired horizontally from the top of a vertical cliff, at a height of \(h\) metres above the sea. It hits the sea 4 seconds after being fired, at a distance of 1000 metres from the base of the cliff, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-4_353_901_479_571}

1. Find the initial speed of the bullet.
2. \(\quad\) Find \(h\).
3. Find the speed of the bullet when it hits the sea.
4. Find the angle between the velocity of the bullet and the horizontal when it hits the sea.

6 In a scene from an action movie, a car is driven off the edge of a cliff and lands on the deck of a boat in the sea, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-4_355_1406_427_324} To land on the boat, the car must move 20 metres horizontally from the cliff. The level of the deck of the boat is 8 metres below the top of the cliff. Assume that the car is a particle which is travelling horizontally when it leaves the top of the cliff and that the car is not affected by air resistance as it moves.

1. Find the time that it takes for the car to reach the deck of the boat.
2. Find the speed at which the car is travelling when it leaves the top of the cliff.
3. Find the speed of the car when it hits the deck of the boat.

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 .

6 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.

4 A boy throws a small ball at a vertical wall. The ball is thrown horizontally, from a point \(O\), at a speed of \(14.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it hits the wall at a point which is 0.2 m below the level of \(O\).

1. Find the horizontal distance from \(O\) to the wall. The boy now moves so that he is 6 m from the wall. He throws the ball at an angle of \(15 ^ { \circ }\) above the horizontal. The ball again hits the wall at a point which is 0.2 m below the level from which it was thrown.
2. Find the speed at which the ball was thrown.

7 A ball is projected horizontally with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at a height of 5 metres above horizontal ground. When the ball has travelled a horizontal distance of 15 metres, it hits the ground. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_433_1296_1674_338}

1. Show that the time it takes for the ball to travel to the point where it hits the ground is 1.01 seconds, correct to three significant figures.
2. Find \(V\).
3. Find the speed of the ball when it hits the ground.
4. Find the angle between the velocity of the ball and the horizontal when the ball hits the ground. Give your answer to the nearest degree.
5. State two assumptions that you have made about the ball while it is moving.

2 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0428f2f2-12c4-4e89-93ab-8cfe2c5aca4a-02_757_889_1482_251}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.

7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.

1. Find the initial speed of the particle.
2. Find the magnitude and direction of the velocity of the particle when it hits the ground.

An aeroplane, travelling horizontally at a speed of 55 ms\(^{-1}\) at a height of 600 metres above horizontal ground, drops a sealed packet of leaflets. Find

1. the time taken by the packet to reach the ground, [3 marks]
2. the horizontal distance moved by the packet during this time. [2 marks]

The packet will split open if it hits the ground at a speed in excess of 125 ms\(^{-1}\).

1. Determine, with explanation, whether the packet will split open. [5 marks]
2. Find the lowest speed at which the aeroplane could be travelling, at the same height of 600 m, to ensure that the packet will split open when it hits the ground. [3 marks]

One of the leaflets is stuck to the front of the packet and becomes detached as it leaves the aeroplane.

1. If the leaflet is modelled as a particle, state how long it takes to reach the ground. [1 mark]
2. Comment on the model of the leaflet as a particle. [2 marks]

A particle is projected horizontally with a speed of 6 m 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. [6]

A particle is projected horizontally with a speed of \(7 \text{ ms}^{-1}\) 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. [6]

A tennis ball is served horizontally at a speed of 24 m s\(^{-1}\) from a height of 2.45 m above the ground.

1. Show that it will clear the net at a point where the net is 1 m high and 12 m from the server. [5]
2. How far beyond the net will it land? [4]