Projectile clearing obstacle

7 A small ball \(B\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(41 ^ { \circ }\) above the horizontal from a point \(O\) which is 1.6 m above horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward 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 show that the equation of the trajectory of \(B\) is $$y = 0.869 x - 0.0390 x ^ { 2 }$$ where the coefficients are correct to 3 significant figures. A vertical fence is 1.5 m from \(O\) and perpendicular to the plane in which \(B\) moves. \(B\) just passes over the fence and subsequently strikes the ground at the point \(A\).
2. Calculate the height of the fence, and the distance from the fence to \(A\).

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 \(d\).

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).

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 \(d\).

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 At a school fair, there is a competition in which people try to kick a football so that it lands in a large rectangular box. The height of the top of the box is 1 metre and its width is 3 metres. One student kicks a football so that it initially moves at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. It hits the top front edge of the box, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-16_465_1342_625_351} Model the football as a particle that is not subject to any resistance forces as it moves.

1. Find the time taken for the football to move from the point where it was kicked to the box.
2. Find the horizontal distance from the point where the football is kicked to the front of the box.
3. If the same student kicks the football at the same angle from the same initial position, what is the speed at which the student should kick the football if it is to hit the top back edge of the box?
4. Explain the significance of modelling the football as a particle in this context.
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3 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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).

1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) 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 \(d\).

6. \begin{figure}[h] \end{figure} A football player strikes a ball giving it an initial speed of \(14 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal as shown in Figure 2. At the instant he strikes the ball it is 0.6 m vertically above the point \(P\) on the ground. The trajectory of the ball is in a vertical plane containing \(P\) and \(M\), the middle of the goal-line. The distance between \(P\) and \(M\) is 12 m and the ground is horizontal. Given that the ball passes over the point \(M\) without bouncing,

1. find, to the nearest degree, the minimum value of \(\alpha\). Given that the crossbar of the goal is 2.4 m above \(M\) and that \(\tan \alpha = \frac { 4 } { 3 }\),
2. show that the ball passes 4.2 m vertically above the crossbar.

1. The fixed points \(X\) and \(Y\) lie on horizontal ground.

At time \(t = 0\), a particle \(P\) is projected from \(X\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at angle \(\theta\) to the ground. Particle \(P\) moves freely under gravity and first hits the ground at \(Y\).

1. Show that \(X Y = \frac { u ^ { 2 } \sin 2 \theta } { g }\) The points \(A\) and \(B\) lie on horizontal ground. A vertical pole \(C D\) has length 5 m .
   The end \(C\) is fixed to the ground between \(A\) and \(B\), where \(A C = 12 \mathrm {~m}\).
   At time \(t = 0\), a particle \(Q\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the ground.
   Particle \(Q\) moves freely under gravity, passes over the pole and first hits the ground at \(B\), as shown in Figure 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-14_335_1179_1032_443} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure}
2. Find the distance \(C B\).
3. Find the height of \(Q\) above \(D\) at the instant when \(Q\) passes over the pole.

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]

During its flight, \(P\) must clear an obstacle of height \(h\) m that is at a horizontal distance of \(32\) m from the point of projection. When \(u = 40\sqrt{2}\) m s\(^{-1}\), \(P\) just clears the obstacle. When \(u = 40\) m s\(^{-1}\), \(P\) only achieves \(80\%\) of the height required to clear the obstacle.

1. Find the two possible values of \(h\). [6]

\includegraphics{figure_3} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of \(30°\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.

1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). [3]
2. Show that \(C\) is approximately 0.63 m below \(T\). [4]

The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining \(30°\). This time \(B\) hits \(T\).

1. Calculate the value of \(V\). [6]
2. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. [1]

\includegraphics{figure_2} Fig. 7 shows a platform \(10\) m long and \(2\) m high standing on horizontal ground. A small ball projected from the surface of the platform at one end, O, just misses the other end, P. The ball is projected at \(68.5°\) to the horizontal with a speed of \(U\text{ms}^{-1}\). Air resistance may be neglected. At time \(t\) seconds after projection, the horizontal and vertical displacements of the ball from O are \(x\) m and \(y\) m.

1. Obtain expressions, in terms of \(U\) and \(t\), for
   1. \(x\),
   2. \(y\). [3]
2. The ball takes \(T\) s to travel from O to P. Show that \(T = \frac{U\sin 68.5°}{4.9}\) and write down a second equation connecting \(U\) and \(T\). [4]
3. Hence show that \(U = 12.0\) (correct to three significant figures). [3]
4. Calculate the horizontal distance of the ball from the platform when the ball lands on the ground. [5]
5. Use the expressions you found in part (i) to show that the cartesian equation of the trajectory of the ball in terms of \(U\) is $$y = x\tan 68.5° - \frac{4.9x^2}{U^2(\cos 68.5°)^2}.$$ Use this equation to show again that \(U = 12.0\) (correct to three significant figures). [4]

In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of 8 ms\(^{-1}\) in a direction making an angle of 30° with the horizontal. \includegraphics{figure_8}

1. Find the time taken by the ball to travel 6 m horizontally. [2 marks]
2. Showing your method clearly, decide whether or not the ball will hit the coconut. [4 marks]
3. Find the greatest height reached by the ball above the level from which it was thrown. [4 marks]
4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of 8 m s\(^{-1}\). [3 marks]
5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut. [2 marks]

A golf ball is hit with initial velocity \(u\) ms\(^{-1}\) at an angle of \(45°\) above the horizontal. The ball passes over a building which is \(15\) m tall at a distance of \(30\) m horizontally from the point where the ball was hit.

1. Find the smallest possible value of \(u\). [7 marks]

When \(u\) has this minimum value,

1. show that the ball does not rise higher than the top of the building. [4 marks]
2. Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
3. Briefly describe two modelling assumptions that you have made. [2 marks]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

A small ball \(B\) moves in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertically upwards axis \(Oy\). \(B\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U \mathrm{m s}^{-1}\) and \(V \mathrm{m s}^{-1}\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.

1. Show that the path of \(B\) has equation \(2U^2 y = 2UVx - gx^2\). [3]

During its motion, \(B\) just clears a vertical wall of height \(\frac{1}{3}a\) m at a horizontal distance \(a\) m from \(O\). \(B\) strikes the ground at a horizontal distance \(3a\) m beyond the wall.

1. Determine the angle of projection of \(B\). Give your answer in degrees correct to 3 significant figures. [5]
2. Given that the speed of projection of \(B\) is \(54.6 \mathrm{m s}^{-1}\), determine the value of \(a\). [2]
3. Hence find the maximum height of \(B\) above the ground during its motion. [3]
4. State one refinement of the model, other than including air resistance, that would make it more realistic. [1]

A tennis ball is projected with velocity vector \((30\mathbf{i} - 14\mathbf{j})\) ms\(^{-1}\) from a point \(P\) which is at a height of \(2.4\) m vertically above a horizontal tennis court. The ball then passes over a net of height \(0.9\) m, before hitting the ground after \(\frac{4}{7}\) s. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The origin \(O\) lies on the ground directly below the point \(P\). The base of the net is \(x\) m from \(O\). \includegraphics{figure_10}

1. Find the speed of the ball when it first hits the ground, giving your answer correct to one decimal place. [3]
2. After \(\frac{2}{5}\) s, the ball is directly above the net.
   1. Find the position vector of the ball after \(\frac{2}{5}\) s.
   2. Hence determine the value of \(x\) and show that the ball clears the net by approximately \(16\) cm. [4]
3. In fact, the ball clears the net by only \(4\) cm.
   1. Explain why the observed value is different from the value calculated in (b)(ii).
   2. Suggest a possible improvement to this model. [2]

A girl is practising netball. She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop. The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.

• The initial velocity of the ball has magnitude \(U\) m s\(^{-1}\).
• The angle of projection is \(40°\).
• The ball is modelled as a particle.
• The hoop is modelled as a point.

This is shown on the diagram below. \includegraphics{figure_12}

1. For \(U = 10\), find
   1. the greatest height above the ground reached by the ball [5]
   2. the distance between the ball and the hoop when the ball is vertically above the hoop. [4]
2. Calculate the value of \(U\) which allows her to hit the hoop. [3]
3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl? [1]
4. Suggest one improvement that might be made to this model. [1]

\includegraphics{figure_11} A projectile is fired from a point \(O\) in a horizontal plane, with initial speed \(V\), at an angle \(\theta\) to the horizontal (see diagram).

1. Show that the range of the projectile on the horizontal plane is $$\frac{2V^2 \sin \theta \cos \theta}{g}.$$ [4]

There are two vertical walls, each of height \(h\), at distances 30 m and 70 m, respectively, from \(O\) with bases on the horizontal plane. The value of \(\theta\) is \(45°\).

1. If the projectile just clears both walls, state the range of the projectile. [1]
2. Hence find the value of \(V\) and of \(h\). [5]