3.02i Projectile motion: constant acceleration model

6. A particle \(P\) is projected from a point \(A\) on horizontal ground with speed \(u\) at an angle of elevation \(\alpha\) and moves freely under gravity. \(P\) hits the ground at the point \(B\).

1. Show that \(A B = \frac { u ^ { 2 } } { g } \sin 2 \alpha\). An archer fires an arrow with an initial speed of \(45 \mathrm {~ms} ^ { - 1 }\) at a target which is level with the point of projection and at a distance of 80 m . Given that the arrow hits the target,
2. find in degrees, correct to 1 decimal place, the two possible angles of projection.
3. Write down, with a reason, which of the two possible angles of projection would give the shortest time of flight.
   (2 marks)
4. Show that the minimum time of flight is 1.8 seconds, correct to 1 decimal place.
   (2 marks)

5. A firework company is testing its new brand of firework, the Sputnik Special. One of the company's employees lights a Sputnik Special on a large area of horizontal ground and it takes off at a small angle to the vertical. After a flight lasting 8 seconds it lands at a distance of 24 metres from the point where it was launched. The employee models the firework as a particle and ignores air resistance and any loss of mass which the Sputnik Special experiences. Using this model, find for this flight of the Sputnik Special,

1. the horizontal and vertical components of the initial velocity,
2. the initial speed, correct to 3 significant figures,
3. the maximum height attained.
4. Comment on the suitability of the modelling assumptions made by the employee.

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.

5. \begin{figure}[h] \end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of \(22 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 7 } { 8 }\). When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.

1. Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
2. Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
3. find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.

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.

7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.

1. 1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
   2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.

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.

7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.

1. 1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
   2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.

5 A ball is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) above the horizontal so as to hit a point \(P\) on a wall. The ball travels in a vertical plane through the point of projection. During the motion, the horizontal and upward vertical displacements of the ball from the point of projection are \(x\) metres and \(y\) metres respectively.

1. Show that, during the flight, the equation of the trajectory of the ball is given by $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$$
2. The ball is projected from a point 1 metre vertically below and \(R\) metres horizontally from the point \(P\).
   1. By taking \(g = 10 \mathrm {~ms} ^ { - 2 }\), show that \(R\) satisfies the equation $$5 R ^ { 2 } \tan ^ { 2 } \alpha - u ^ { 2 } R \tan \alpha + 5 R ^ { 2 } + u ^ { 2 } = 0$$
   2. Hence, given that \(u\) and \(R\) are constants, show that, for \(\tan \alpha\) to have real values, \(R\) must satisfy the inequality $$R ^ { 2 } \leqslant \frac { u ^ { 2 } \left( u ^ { 2 } - 20 \right) } { 100 }$$
   3. Given that \(R = 5\), determine the minimum possible speed of projection.

7 A particle is projected from a point on a plane which is inclined at an angle \(\alpha\) to the horizontal. The particle is projected up the plane with velocity \(u\) at an angle \(\theta\) above the plane. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-5_401_748_516_644}

1. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), show that the range up the plane is $$\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
2. Hence, using the identity \(2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )\), show that, as \(\theta\) varies, the range up the plane is a maximum when \(\theta = \frac { \pi } { 4 } - \frac { \alpha } { 2 }\).
3. Given that the particle strikes the plane at right angles, show that $$2 \tan \theta = \cot \alpha$$

5 A boy throws a small ball from a height of 1.5 m above horizontal ground with initial velocity \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The ball hits a small can placed on a vertical wall of height 2.5 m , which is at a horizontal distance of 5 m from the initial position of the ball, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-3_499_1180_1283_424}

1. Show that \(\alpha\) satisfies the equation $$49 \tan ^ { 2 } \alpha - 200 \tan \alpha + 89 = 0$$
2. Find the two possible values of \(\alpha\), giving your answers to the nearest \(0.1 ^ { \circ }\).
3. 1. To knock the can off the wall, the horizontal component of the velocity of the ball must be greater than \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that, for one of the possible values of \(\alpha\) found in part (b), the can will be knocked off the wall, and for the other, it will not be knocked off the wall.
      (3 marks)
   2. Given that the can is knocked off the wall, find the direction in which the ball is moving as it hits the can.

7 A projectile is fired with speed \(u\) from a point \(O\) on a plane which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired at an angle \(\theta\) to the inclined plane and moves in a vertical plane through a line of greatest slope of the inclined plane. The projectile lands at a point \(P\), lower down the inclined plane, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-5_415_1098_495_463}

1. Find, in terms of \(u , g , \theta\) and \(\alpha\), the greatest perpendicular distance of the projectile from the plane.
2. 1. Find, in terms of \(u , g , \theta\) and \(\alpha\), the time of flight from \(O\) to \(P\).
   2. By using the identity \(\cos A \cos B + \sin A \sin B = \cos ( A - B )\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta - \alpha ) } { g \cos ^ { 2 } \alpha }\).
   3. Hence, by using the identity \(2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )\) or otherwise, show that, as \(\theta\) varies, the maximum possible distance \(O P\) is \(\frac { u ^ { 2 } } { g ( 1 - \sin \alpha ) }\).
      (5 marks)

2 A projectile is fired from a point \(O\) on top of a hill with initial velocity \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal and moves in a vertical plane. The horizontal and upward vertical distances of the projectile from \(O\) are \(x\) metres and \(y\) metres respectively.

1. 1. Show that, during the flight, the equation of the trajectory of the projectile is given by $$y = x \tan \theta - \frac { g x ^ { 2 } } { 12800 } \left( 1 + \tan ^ { 2 } \theta \right)$$
   2. The projectile hits a target \(A\), which is 20 m vertically below \(O\) and 400 m horizontally from \(O\). \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-04_392_1031_970_460} Show that $$49 \tan ^ { 2 } \theta - 160 \tan \theta + 41 = 0$$
2. 1. Find the two possible values of \(\theta\). Give your answers to the nearest \(0.1 ^ { \circ }\).
   2. Hence find the shortest possible time of the flight of the projectile from \(O\) to \(A\).
3. State a necessary modelling assumption for answering part (a)(i).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-05_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-07_2484_1709_223_153}

3 (In this question, use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A golf ball is hit from a point \(O\) on a horizontal golf course with a velocity of \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The golf ball travels in a vertical plane through \(O\). During its flight, the horizontal and upward vertical distances of the golf ball from \(O\) are \(x\) and \(y\) metres respectively.

1. Show that the equation of the trajectory of the golf ball during its flight is given by $$x ^ { 2 } \tan ^ { 2 } \theta - 320 x \tan \theta + \left( x ^ { 2 } + 320 y \right) = 0$$
2. 1. The golf ball hits the top of a tree, which has a vertical height of 8 m and is at a horizontal distance of 150 m from \(O\). Find the two possible values of \(\theta\).
   2. Which value of \(\theta\) gives the shortest possible time for the golf ball to travel from \(O\) to the top of the tree? Give a reason for your choice of \(\theta\).

6 A projectile is fired from a point \(O\) on a plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The projectile is fired up the plane with velocity \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the inclined plane. The projectile travels in a vertical plane containing a line of greatest slope of the inclined plane. The projectile hits a target \(T\) on the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-16_481_922_664_593}

1. Given that \(O T = 200 \mathrm {~m}\), determine the value of \(u\).
2. Find the greatest perpendicular distance of the projectile from the inclined plane.
   (4 marks)
   
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-18_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-19_2486_1714_221_153}

3 (In this question, take \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A projectile is fired from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal so as to pass through a point \(P\). The projectile travels in a vertical plane through \(O\) and \(P\). The point \(P\) is at a horizontal distance \(2 k\) from \(O\) and at a vertical distance \(k\) above \(O\).

1. Show that \(\alpha\) satisfies the equation $$20 k \tan ^ { 2 } \alpha - 2 u ^ { 2 } \tan \alpha + u ^ { 2 } + 20 k = 0$$
2. Deduce that $$u ^ { 4 } - 20 k u ^ { 2 } - 400 k ^ { 2 } \geqslant 0$$

5 A particle is projected from a point \(O\) on a smooth plane, which is inclined at \(25 ^ { \circ }\) to the horizontal. The particle is projected up the plane with velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(30 ^ { \circ }\) above the plane. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-12_518_839_552_630}

1. Find the time taken by the particle to travel from \(O\) to \(A\).
2. The coefficient of restitution between the particle and the inclined plane is \(\frac { 2 } { 3 }\). Find the speed of the particle as it rebounds from the inclined plane at \(A\). (8 marks)

3 A player projects a basketball with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The basketball travels in a vertical plane through the point of projection and goes into the basket. During the motion, the horizontal and upward vertical displacements of the basketball from the point of projection are \(x\) metres and \(y\) metres respectively. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-06_737_937_513_550}

1. Find an expression for \(y\) in terms of \(x , u , g\) and \(\tan \theta\).
2. The player projects the basketball with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.5 metres vertically below and 5 metres horizontally from the basket.
   1. Show that the two possible values of \(\theta\) are approximately \(63.1 ^ { \circ }\) and \(32.6 ^ { \circ }\), correct to three significant figures.
   2. Given that the player projects the basketball at \(63.1 ^ { \circ }\) to the horizontal, find the direction of the motion of the basketball as it enters the basket. Give your answer to the nearest degree.
3. State a modelling assumption needed for answering parts (a) and (b) of this question.
   (1 mark)

5 A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected down the plane with velocity \(u\) at an angle \(\alpha\) above the plane. The particle first strikes the plane at a point \(P\), as shown in the diagram. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-12_389_789_557_639}

1. Given that the time of flight from \(O\) to \(P\) is \(T\), find an expression for \(u\) in terms of \(\theta , \alpha , T\) and \(g\).
2. Using the identity \(\cos ( X - Y ) = \cos X \cos Y + \sin X \sin Y\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \alpha \cos ( \alpha - \theta ) } { g \cos ^ { 2 } \theta }\).
   (6 marks)

5 A small smooth ball is dropped from a height of \(h\) above a point \(A\) on a fixed smooth plane inclined at an angle \(\theta\) to the horizontal. The ball falls vertically and collides with the plane at the point \(A\). The ball rebounds and strikes the plane again at a point \(B\), as shown in the diagram. The points \(A\) and \(B\) lie on a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-12_318_636_548_712}

1. Explain whether or not the component of the velocity of the ball parallel to the plane is changed by the collision.
2. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(h , \theta , e\) and \(g\), the components of the velocity of the ball parallel to and perpendicular to the plane immediately after the collision.
3. Show that the distance \(A B\) is given by $$4 h e ( e + 1 ) \sin \theta$$

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]

6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h] \end{figure}

1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
3. Find the value of \(\cos \theta\) correct to 3 significant figures.
4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.

4 A plane is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. A particle is projected from a point A on the plane with speed \(V \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(\phi ^ { \circ }\) with a line of greatest slope of the plane. The particle lands at a point B on the plane, as shown in the diagram, and the time of flight is \(T\) seconds. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-4_332_872_461_246} \begin{enumerate}[label=(\alph*)] \item By considering the motion of the particle perpendicular to the plane, show that \(\mathrm { T } = \frac { 2 \mathrm {~V} \sin \phi } { \mathrm {~g} \cos \theta }\). Consider the case when \(\theta = 30 , \phi = 25\) and \(V = 20\). \item

1. Calculate the distance AB .
2. State, with reasons but without any detailed calculations, what effect each of the following actions would have on the distance AB .

6 A section of a golf practice ground is inclined at \(15 ^ { \circ }\) to the horizontal. A golfer is hitting a ball up and down a line of greatest slope of this section of the practice ground. The golfer hits the ball up the slope, so that the ball initially makes an angle of \(30 ^ { \circ }\) with the slope. The ball first bounces on the slope 50 m from its point of projection.

1. Determine the initial speed of the ball. The golfer now hits the ball down the slope. The ball initially moves with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the ball initially travels at an angle \(\theta\) above the horizontal, as shown in Fig. 6. The ball first bounces at a point a distance \(L\) m down the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-6_545_791_794_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Show that \(\mathrm { L } = \frac { 800 } { \mathrm {~g} } \left( \frac { \sin \theta \cos \theta } { \cos 15 ^ { \circ } } + \frac { \sin 15 ^ { \circ } \cos ^ { 2 } \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\). You are given that \(\frac { \mathrm { dL } } { \mathrm { d } \theta } = \frac { 800 } { \mathrm {~g} } \left( \frac { \cos 2 \theta } { \cos 15 ^ { \circ } } - \frac { \sin 15 ^ { \circ } \sin 2 \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\).
3. Determine the value of \(\theta\) for which \(\frac { \mathrm { d } L } { \mathrm {~d} \theta } = 0\).
4. Hence determine the maximum distance the golfer can hit the ball down the slope.

7 A plane is inclined at \(30 ^ { \circ }\) above the horizontal. A particle is projected up the plane from a point C on the plane with a velocity of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above a line of greatest slope of the plane. The particle hits the plane at D. See Fig. 7. \begin{figure}[h] \end{figure}

1. Using the standard model for projectile motion, show that the time of flight, \(T\), is given by $$T = \frac { 28 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
2. Calculate the distance CD. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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