Projectile on inclined plane

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.

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.

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 $$2 \tan \theta = \cot \alpha$$

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)

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}

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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9 A small ball P is projected with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(( \alpha + \theta )\) from a point O at the bottom of a plane inclined at \(\alpha\) to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}

1. By deriving an expression, in terms of \(\theta\), \(\alpha\) and \(g\), for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
2. By using the identity \(2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )\), determine, in terms of \(g\) and \(\sin \alpha\), an expression for the maximum range of P up the plane, as \(\theta\) varies.
3. Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length \(b\), is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius \(a\) and length \(\frac { 1 } { 2 } a \pi\). The centre of the arc is O , which is vertically above B . A bead P of mass \(m\) is threaded on the wire and projected from B with speed \(u\) towards C . The angle BOP when P is between B and C is denoted by \(\theta\), as shown in the diagram.

7 A particle is projected from a point \(O\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is projected down the plane with velocity \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the plane and first strikes it 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]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}

1. Show that the time taken by the particle to travel from \(O\) to \(A\) is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
2. Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at \(A\).
3. The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.

3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.

1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.

12 A projectile is launched from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right)$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

12 Points \(A\) and \(B\) lie on a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal, with \(B\) above \(A\). A particle is projected from \(A\) with speed \(u\) at an angle \(\theta\) to the plane and subsequently strikes the plane at right angles at \(B\).

1. Show that \(2 \tan \alpha \tan \theta = 1\).
2. In either order, show that
   1. the vertical height of \(B\) above \(A\) is \(\frac { 2 u ^ { 2 } \tan ^ { 2 } \alpha } { g \left( 1 + 4 \tan ^ { 2 } \alpha \right) }\),
   2. the time of flight from \(A\) to \(B\) is \(\frac { 2 u \sec \alpha } { g \sqrt { 1 + 4 \tan ^ { 2 } \alpha } }\).

13 A cricket ball is hit from a point \(P\) on a sloping field. The initial velocity of the ball is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above the field, which under the path of the ball slopes upwards at \(10 ^ { \circ }\) to the horizontal. Air resistance is to be ignored.

1. Find the vertical height of the ball above the field after 2.5 seconds.
2. The ball lands on the field at the point \(X\). Find the distance \(P X\).

12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

12 A particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

4 A particle is projected with velocity \(V\), at an angle of elevation of \(60 ^ { \circ }\) to the horizontal, from a point on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the particle is in a vertical plane through a line of greatest slope. If \(R _ { 1 }\) and \(R _ { 2 }\) are the respective ranges when the particle is projected up the plane and down the plane, show that $$R _ { 2 } = 2 R _ { 1 }$$

\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\text{ s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\text{ m}\) and \(y\text{ m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]

A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}

1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
2. Find the distance \(PQ\). [7 marks]

A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]

In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}

1. Determine the following, in either order.
   [9]

After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.

1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]

A particle is projected with velocity \(V\) at an angle \(\alpha\) to the horizontal up a plane inclined at \(\beta\) to the horizontal, where \(\alpha > \beta\).

1. Show that the time of flight is \(\frac{2V \sin(\alpha - \beta)}{g \cos \beta}\). [3]
2. Show that the range on the inclined plane is \(\frac{2V^2 \sin(\alpha - \beta) \cos \alpha}{g \cos^2 \beta}\). [4]
3. If the particle strikes the plane at right angles, prove that \(\tan \alpha = \cot \beta + 2 \tan \beta\). [5]