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Edexcel M2 2012 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-12_602_1175_237_386} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.
  1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed of projection.
  3. Find the time after projection when the stone is moving parallel to \(O B\).
Edexcel M2 2013 June Q1
7 marks Moderate -0.3
  1. A caravan of mass 600 kg is towed by a car of mass 900 kg along a straight horizontal road. The towbar joining the car to the caravan is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having magnitude 300 N . The total resistance to motion of the caravan is modelled as having magnitude 150 N . At a given instant the car and the caravan are moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the power being developed by the car's engine at this instant.
    2. Find the tension in the towbar at this instant.
    3. A ball of mass 0.2 kg is projected vertically upwards from a point \(O\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance acting on the ball is modelled as a force of constant magnitude 1.24 N and the ball is modelled as a particle. Find, using the work-energy principle, the speed of the ball when it first reaches the point which is 8 m vertically above \(O\).
      (6)
    4. A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is given by
    $$v = \frac { 1 } { 2 } t ^ { 2 } - 3 t + 4$$ Find
  2. the times when \(P\) is at rest,
  3. the total distance travelled by \(P\) between \(t = 0\) and \(t = 4\).
Edexcel M2 2013 June Q4
11 marks Standard +0.8
  1. A rough circular cylinder of radius \(4 a\) is fixed to a rough horizontal plane with its axis horizontal. A uniform rod \(A B\), of weight \(W\) and length \(6 a \sqrt { 3 }\), rests with its lower end \(A\) on the plane and a point \(C\) of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-06_389_862_482_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that \(A C = 4 a \sqrt { } 3\) The coefficient of friction between the rod and the cylinder is \(\frac { \sqrt { } 3 } { 3 }\) and the coefficient of friction between the rod and the plane is \(\mu\). Given that friction is limiting at both \(A\) and \(C\),
  2. find the value of \(\mu\).
Edexcel M2 2013 June Q5
13 marks Standard +0.3
5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(m\) respectively, are on a smooth horizontal table. Particle \(Q\) is at rest and particle \(P\) collides directly with it when moving with speed \(u\). After the collision the total kinetic energy of the two particles is \(\frac { 3 } { 4 } m u ^ { 2 }\). Find
  1. the speed of \(Q\) immediately after the collision,
  2. the coefficient of restitution between the particles.
Edexcel M2 2013 June Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_442_871_264_525} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform triangular lamina \(A B C\) of mass \(M\) is such that \(A B = A C , B C = 2 a\) and the distance of \(A\) from \(B C\) is \(h\). A line, parallel to \(B C\) and at a distance \(\frac { 2 h } { 3 }\) from \(A\), cuts \(A B\) at \(D\) and cuts \(A C\) at \(E\), as shown in Figure 2.
It is given that the mass of the trapezium \(B C E D\) is \(\frac { 5 M } { 9 }\).
  1. Show that the centre of mass of the trapezium \(B C E D\) is \(\frac { 7 h } { 45 }\) from \(B C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_357_679_1354_630} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The portion \(A D E\) of the lamina is folded through \(180 ^ { \circ }\) about \(D E\) to form the folded lamina shown in Figure 3.
  2. Find the distance of the centre of mass of the folded lamina from \(B C\). The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(D E\) and the downward vertical is \(\alpha\).
  3. Find \(\tan \alpha\) in terms of \(a\) and \(h\).
Edexcel M2 2013 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-13_520_1027_296_447} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9 a\) from \(O\) and at a height \(6 a\) above the level of \(O\). The ball is projected with speed \(\sqrt { } ( 27 a g )\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
  1. Show that \(\tan ^ { 2 } \theta - 6 \tan \theta + 5 = 0\) The two possible angles of projection are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), where \(\theta _ { 1 } > \theta _ { 2 }\).
  2. Find \(\tan \theta _ { 1 }\) and \(\tan \theta _ { 2 }\). The particle is projected at the larger angle \(\theta _ { 1 }\).
  3. Show that the time of flight from \(O\) to \(T\) is \(\sqrt { } \left( \frac { 78 a } { g } \right)\).
  4. Find the speed of the particle immediately before it hits \(T\).
Edexcel M2 2013 June Q1
8 marks Moderate -0.3
  1. Three particles of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are positioned at the points with coordinates \(( a , 3 ) , ( 3 , - 1 )\) and \(( - 2,4 )\) respectively. Given that the centre of mass of the particles is at the point with coordinates \(( 0,2 )\), find
    1. the value of \(m\),
    2. the value of \(a\).
    3. A car has mass 1200 kg . The maximum power of the car's engine is 32 kW . The resistance to motion due to non-gravitational forces is modelled as a force of constant magnitude 800 N . When the car is travelling on a horizontal road at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine of the car is working at maximum power.
    4. Find the value of \(V\).
    The car now travels downhill on a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The resistance to motion due to non-gravitational forces is still modelled as a force of constant magnitude 800 N . Given that the engine of the car is again working at maximum power,
  2. find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2013 June Q3
13 marks Standard +0.3
3 A particle \(P\) of mass 0.25 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = ( 2 - 4 t ) \mathbf { i } + \left( t ^ { 2 } + 2 t \right) \mathbf { j }$$ When \(t = 0 , P\) is at the point with position vector ( \(2 \mathbf { i } - 4 \mathbf { j }\) ) m with respect to a fixed origin \(O\). When \(t = 3 , P\) is at the point \(A\). Find
  1. the momentum of \(P\) when \(t = 3\),
  2. the magnitude of \(\mathbf { F }\) when \(t = 3\),
  3. the position vector of \(A\).
Edexcel M2 2013 June Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-5_496_1264_316_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(O\) and \(B\) are on horizontal ground. The point \(A\) is \(h\) metres vertically above \(O\). A particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha ^ { \circ }\) to the horizontal. The particle moves freely under gravity and hits the ground at \(B\), as shown in Figure 1. The speed of \(P\) immediately before it hits the ground is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the value of \(h\). Given that 1.5 s after it is projected from \(A , P\) is at a point 4 m above the level of \(A\), find
  2. the value of \(\alpha\),
  3. the direction of motion of \(P\) immediately before it reaches \(B\).
Edexcel M2 2013 June Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-6_501_696_316_726} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform L-shaped lamina \(O A B C D E\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from
  1. \(O E\),
  2. \(O A\). The lamina is freely suspended from \(O\) and hangs in equilibrium with \(O E\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac { 4 } { 3 }\).
  3. Find the value of \(a\).
Edexcel M2 2013 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-7_574_1257_214_448} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\) has weight 30 N and length 3 m . The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at \(15 ^ { \circ }\) to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.
  1. Show that the normal reaction at \(P\) has magnitude 25 N .
  2. Find the magnitude of the force on the rod at \(A\). The coefficient of friction between the rod and the peg is \(\mu\).
  3. Find the range of possible values of \(\mu\).
Edexcel M2 2013 June Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-8_453_839_219_649} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2 m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).
  1. Show that \(v = 2 w\). The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.
  2. Find the coefficient of restitution between \(P\) and \(Q\).
Edexcel M2 2013 June Q1
5 marks Moderate -0.8
  1. A particle \(P\) of mass 2 kg is moving with velocity \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 3 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\).
Find the speed of \(P\) immediately after the impulse is applied.
(5)
Edexcel M2 2013 June Q2
7 marks Standard +0.3
2. A particle \(P\) of mass 3 kg moves from point \(A\) to point \(B\) up a line of greatest slope of a fixed rough plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 Given that \(A B = 15 \mathrm {~m}\) and that the speed of \(P\) at \(A\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find
  1. the work done against friction as \(P\) moves from \(A\) to \(B\),
  2. the speed of \(P\) at \(B\).
Edexcel M2 2013 June Q3
13 marks Moderate -0.8
3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find
  1. the times when \(P\) is instantaneously at rest,
  2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
  3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
Edexcel M2 2013 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-06_736_725_258_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C D E F\) is a regular hexagon with centre \(O\) and sides of length 2 m , as shown in Figure 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-06_574_723_1288_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The triangles \(O A F\) and \(O E F\) are removed to form the uniform lamina \(O A B C D E\), shown in Figure 2.
  1. Find the distance of the centre of mass of \(O A B C D E\) from \(O\). The lamina \(O A B C D E\) is freely suspended from \(E\) and hangs in equilibrium.
  2. Find the size of the angle between \(E O\) and the downward vertical.
Edexcel M2 2013 June Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-09_522_997_276_477} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(m\) is attached to the rod at \(B\). The rod is held in equilibrium at an angle \(\theta\) to the horizontal by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = b\), as shown in Figure 3. The force at \(C\) acts at right angles to \(A B\) and in the vertical plane containing \(A B\).
  1. Show that \(F = \frac { 3 a m g \cos \theta } { b }\).
  2. Find, in terms of \(a , b , g , m\) and \(\theta\),
    1. the horizontal component of the force acting on the rod at \(A\),
    2. the vertical component of the force acting on the rod at \(A\). Given that the force acting on the rod at \(A\) acts along the rod,
  3. find the value of \(\frac { a } { b }\).
Edexcel M2 2013 June Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-11_694_1004_264_529} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A ball is projected from a point \(A\) which is 8 m above horizontal ground as shown in Figure 4. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\). The speed of the ball immediately before it hits the ground is \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the value of \(u\). The time taken for the ball to move from \(A\) to \(B\) is 2 seconds. Find
  2. the value of \(\theta\),
  3. the minimum speed of the ball on its path from \(A\) to \(B\).
Edexcel M2 2013 June Q7
15 marks Standard +0.3
7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). The particles \(P , Q\) and \(R\) have masses \(2 m , 3 m\) and \(4 m\) respectively. Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with it. The coefficient of restitution between each pair of particles is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision with \(P\) is \(\frac { 2 } { 5 } ( 1 + e ) u\). After the collision between \(P\) and \(Q\) there is a direct collision between \(Q\) and \(R\).
    Given that \(e = \frac { 3 } { 4 }\), find
    1. the speed of \(Q\) after this collision,
    2. the speed of \(R\) after this collision. Immediately after the collision between \(Q\) and \(R\), the rate of increase of the distance between \(P\) and \(R\) is \(V\).
  3. Find \(V\) in terms of \(u\).
Edexcel M2 2014 June Q1
8 marks Moderate -0.3
  1. A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 16 }\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the van’s engine is working at a constant rate of 25 kW .
    1. Find the value of \(R\).
    The power developed by the van’s engine is now increased to 30 kW . The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
Edexcel M2 2014 June Q2
7 marks Moderate -0.3
  1. A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse \(( - 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) s on the ball. Immediately after receiving the impulse the ball has velocity \(( 12 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find
  1. the speed of the ball immediately before the impact,
  2. the size of the angle through which the direction of motion of the ball is deflected by the impact.
Edexcel M2 2014 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-05_617_604_226_678} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.
  1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
  2. Find the distance \(A G\).
Edexcel M2 2014 June Q4
10 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-07_737_823_223_532} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a lamina \(L\). It is formed by removing a square \(P Q R S\) from a uniform triangle \(A B C\). The triangle \(A B C\) is isosceles with \(A C = B C\) and \(A B = 12 \mathrm {~cm}\). The midpoint of \(A B\) is \(D\) and \(D C = 8 \mathrm {~cm}\). The vertices \(P\) and \(Q\) of the square lie on \(A B\) and \(P Q = 4 \mathrm {~cm}\). The centre of the square is \(O\). The centre of mass of \(L\) is at \(G\).
  1. Find the distance of \(G\) from \(A B\). When \(L\) is freely suspended from \(A\) and hangs in equilibrium, the line \(A B\) is inclined at \(25 ^ { \circ }\) to the vertical.
  2. Find the distance of \(O\) from \(D C\).
Edexcel M2 2014 June Q5
13 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-09_460_974_242_484} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 2 kg is released from rest at a point \(A\) on a rough inclined plane and slides down a line of greatest slope. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) is 5 m from \(A\) on the line of greatest slope through \(A\), as shown in Figure 3.
  1. Find the potential energy lost by \(P\) as it moves from \(A\) to \(B\). The speed of \(P\) as it reaches \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Use the work-energy principle to find the magnitude of the constant frictional force acting on \(P\) as it moves from \(A\) to \(B\).
    2. Find the coefficient of friction between \(P\) and the plane. The particle \(P\) is now placed at \(A\) and projected down the plane towards \(B\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the frictional force remains constant,
  3. find the speed of \(P\) as it reaches \(B\).
Edexcel M2 2014 June Q6
13 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-11_711_917_219_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) is projected from a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 4 } { 5 }\). The point \(A\) is 10 m vertically above the point \(O\) which is on horizontal ground, as shown in Figure 4. The particle \(P\) moves freely under gravity and reaches the ground at the point \(B\). Calculate
  1. the greatest height above the ground of \(P\), as it moves from \(A\) to \(B\),
  2. the distance \(O B\). The point \(C\) lies on the path of \(P\). The direction of motion of \(P\) at \(C\) is perpendicular to the direction of motion of \(P\) at \(A\).
  3. Find the time taken by \(P\) to move from \(A\) to \(C\).