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Edexcel M2 2010 January Q1
8 marks Moderate -0.3
  1. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction, where \(v = 3 t ^ { 2 } - 4 t + 3\). When \(t = 0 , P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity.
    (8)
  2. Two particles, \(P\), of mass \(2 m\), and \(Q\), of mass \(m\), are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between the particles is \(e\), where \(e < 1\). Find, in terms of \(u\) and \(e\),
    1. the speed of \(P\) immediately after the collision,
    2. the speed of \(Q\) immediately after the collision.
    3. A particle of mass 0.5 kg is projected vertically upwards from ground level with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It comes to instantaneous rest at a height of 10 m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\).
      (6)
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8dac5891-0dfd-49b4-ada4-0ecb875cf6aa-05_547_1132_125_396} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The points \(A , B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass 0.25 kg . Immediately before being struck, the ball is moving along the horizontal line \(A B\) with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after being struck, the ball moves along the horizontal line \(B C\) with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(60 ^ { \circ }\) with the original direction of motion \(A B\), as shown in Figure 1. Find, to 3 significant figures,
  3. the magnitude of the impulse given to the ball,
  4. the size of the angle that the direction of this impulse makes with the original direction of motion \(A B\).
Edexcel M2 2010 January Q5
11 marks Moderate -0.3
  1. A cyclist and her bicycle have a total mass of 70 kg . She cycles along a straight horizontal road with constant speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She is working at a constant rate of 490 W .
    1. Find the magnitude of the resistance to motion.
    The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\), at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the nongravitational resistance to motion is modelled as \(40 U\) newtons. She is now working at a constant rate of 24 W .
  2. Find the value of \(U\).
Edexcel M2 2010 January Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8dac5891-0dfd-49b4-ada4-0ecb875cf6aa-09_571_1101_128_416} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\), of mass 20 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is 0.5 . Find the magnitude of the normal reaction of the ground on the rod at \(A\).
Edexcel M2 2010 January Q7
11 marks Standard +0.3
  1. \hspace{0pt} [The centre of mass of a semi-circular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8dac5891-0dfd-49b4-ada4-0ecb875cf6aa-11_656_1274_421_355} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A template \(T\) consists of a uniform plane lamina \(P Q R O S\), as shown in Figure 3. The lamina is bounded by two semicircles, with diameters \(S O\) and \(O R\), and by the sides \(S P , P Q\) and \(Q R\) of the rectangle \(P Q R S\). The point \(O\) is the mid-point of \(S R , P Q = 12 \mathrm {~cm}\) and \(Q R = 2 x \mathrm {~cm}\).
  1. Show that the centre of mass of \(T\) is a distance \(\frac { 4 \left| 2 x ^ { 2 } - 3 \right| } { 8 x + 3 \pi } \mathrm {~cm}\) from \(S R\). The template \(T\) is freely suspended from the point \(P\) and hangs in equilibrium.
    Given that \(x = 2\) and that \(\theta\) is the angle that \(P Q\) makes with the horizontal,
  2. show that \(\tan \theta = \frac { 48 + 9 \pi } { 22 + 6 \pi }\).
Edexcel M2 2010 January Q8
17 marks Standard +0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in a horizontal and upward vertical direction respectively]
A particle \(P\) is projected from a fixed point \(O\) on horizontal ground with velocity \(u ( \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(c\) and \(u\) are positive constants. The particle moves freely under gravity until it strikes the ground at \(A\), where it immediately comes to rest. Relative to \(O\), the position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\).
  1. Show that $$y = c x - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } }$$ Given that \(u = 7 , O A = R \mathrm {~m}\) and the maximum vertical height of \(P\) above the ground is \(H \mathrm {~m}\),
  2. using the result in part (a), or otherwise, find, in terms of \(c\),
    1. \(R\)
    2. \(H\). Given also that when \(P\) is at the point \(Q\), the velocity of \(P\) is at right angles to its initial velocity,
  3. find, in terms of \(c\), the value of \(x\) at \(Q\).
Edexcel M2 2011 January Q1
6 marks Moderate -0.8
  1. A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg . The resistance to motion is modelled as a constant force of magnitude 32 N . The rate at which the cyclist works is 384 W . The cyclist accelerates until he reaches a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find
  1. the value of \(v\),
  2. the acceleration of the cyclist at the instant when the speed is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2011 January Q2
5 marks Moderate -0.3
2. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( - 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { N }\) s. Find the kinetic energy of the particle immediately after receiving the impulse.
(5) \includegraphics[max width=\textwidth, alt={}, center]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-03_41_1571_504_185}
Edexcel M2 2011 January Q3
8 marks Moderate -0.3
3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \(\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. Find
  1. the velocity of the particle at time \(t\) seconds,
  2. the displacement of the particle from the origin at time \(t\) seconds,
  3. the values of \(t\) at which the particle is instantaneously at rest.
Edexcel M2 2011 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-06_365_776_264_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(A B\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\frac { 1 } { 4 }\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,
  1. find the work done in dragging the box from \(A\) to \(B\). The box is released from rest at the point \(B\) and slides down the slope. Using the workenergy principle, or otherwise,
  2. find the speed of the box as it reaches \(A\).
    January 2011
Edexcel M2 2011 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-10_823_908_269_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform L-shaped lamina \(A B C D E F\), shown in Figure 2, has sides \(A B\) and \(F E\) parallel, and sides \(B C\) and \(E D\) parallel. The pairs of parallel sides are 9 cm apart. The points \(A , F\), \(D\) and \(C\) lie on a straight line. \(A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }\), and \(\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }\).
  1. Find the distance of the centre of mass of the lamina from
    1. side \(A B\),
    2. side \(B C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find, to the nearest degree, the size of the angle between \(A B\) and the vertical.
Edexcel M2 2011 January Q6
12 marks Moderate -0.3
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-12_689_1042_360_459} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector 10j metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.
  1. For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf { r }\) metres, of \(P\) at time \(t\) seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
  2. Find the value of \(T\).
  3. Find the velocity of \(P\) at time \(t\) seconds \(( 0 \leqslant t \leqslant T )\). When \(P\) is at the point \(B\), the direction of motion of \(P\) is \(45 ^ { \circ }\) below the horizontal.
  4. Find the time taken for \(P\) to move from \(A\) to \(B\).
  5. Find the speed of \(P\) as it passes through \(B\).
Edexcel M2 2011 January Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-14_442_986_264_479} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).
Edexcel M2 2011 January Q8
13 marks Standard +0.3
  1. A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J . The coefficient of restitution between \(P\) and the wall is \(\frac { 1 } { 3 }\).
    1. Show that \(m = 4\).
      (6)
    After rebounding from the wall, \(P\) collides directly with a particle \(Q\) which is moving towards \(P\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(Q\) is 2 kg and the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\).
  2. Show that there will be a second collision between \(P\) and the wall.
Edexcel M2 2012 January Q1
4 marks Moderate -0.8
  1. A tennis ball of mass 0.1 kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The racquet exerts an impulse of \(( - 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { Ns }\) on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit.
  2. A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - 3 t ^ { 2 } \mathbf { j }\).
Find
  1. the speed of \(P\) when \(t = 4\)
  2. the acceleration of \(P\) when \(t = 4\) Given that \(P\) is at the point with position vector \(( - 4 \mathbf { i } + \mathbf { j } ) \mathrm { m }\) when \(t = 1\),
  3. find the position vector of \(P\) when \(t = 4\)
Edexcel M2 2012 January Q3
10 marks Standard +0.3
3. A cyclist and her cycle have a combined mass of 75 kg . The cyclist is cycling up a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 20 N . At the instant when the cyclist has a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), she is decelerating at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the rate at which the cyclist is working at this instant. When the cyclist passes the point \(A\) her speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 20 N .
  2. Use the work-energy principle to find the distance \(A B\).
Edexcel M2 2012 January Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-06_415_981_237_475} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The trapezium \(A B C D\) is a uniform lamina with \(A B = 4 \mathrm {~m}\) and \(B C = C D = D A = 2 \mathrm {~m}\), as shown in Figure 1.
  1. Show that the centre of mass of the lamina is \(\frac { 4 \sqrt { } 3 } { 9 } \mathrm {~m}\) from \(A B\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
  2. Find the angle between \(D C\) and the vertical through \(D\).
Edexcel M2 2012 January Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-08_597_981_217_461} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has mass 4 kg and length 1.4 m . The end \(A\) is resting on rough horizontal ground. A light string \(B C\) has one end attached to \(B\) and the other end attached to a fixed point \(C\). The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at \(20 ^ { \circ }\) to the ground, as shown in Figure 2.
  1. Find the tension in the string. Given that the rod is about to slip,
  2. find the coefficient of friction between the rod and the ground.
Edexcel M2 2012 January Q6
15 marks Standard +0.3
6. Three identical particles, \(A , B\) and \(C\), lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The mass of each particle is \(m\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\).
  1. Find, in terms of \(u\),
    1. the speed of \(A\) after this collision,
    2. the speed of \(B\) after this collision.
  2. Show that the kinetic energy lost in this collision is \(\frac { 5 } { 36 } m u ^ { 2 }\) After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\).
  3. Find, in terms of \(u\), the speed of \(C\) immediately after this collision between \(B\) and \(C\).
Edexcel M2 2012 January Q7
15 marks Standard +0.3
7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-12_414_1234_338_354} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and passes through the point \(A\) at time \(t\) seconds after projection. The point \(B\) is on the horizontal plane vertically below \(A\), as shown in Figure 3. It is given that \(O B = 2 A B\). Find
  1. the value of \(t\),
  2. the speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the ball at the instant when it passes through \(A\). At another point \(C\) on the path the speed of the ball is also \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the time taken for the ball to travel from \(O\) to \(C\).
Edexcel M2 2013 January Q1
5 marks Standard +0.3
  1. Two uniform rods \(A B\) and \(B C\) are rigidly joined at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). Rod \(A B\) has length 0.5 m and mass 2 kg . Rod \(B C\) has length 2 m and mass 3 kg . The centre of mass of the framework of the two rods is at \(G\).
    1. Find the distance of \(G\) from \(B C\).
    The distance of \(G\) from \(A B\) is 0.6 m .
    The framework is suspended from \(A\) and hangs freely in equilibrium.
  2. Find the angle between \(A B\) and the downward vertical at \(A\).
Edexcel M2 2013 January Q2
9 marks Moderate -0.3
2. A lorry of mass 1800 kg travels along a straight horizontal road. The lorry's engine is working at a constant rate of 30 kW . When the lorry's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The magnitude of the resistance to the motion of the lorry is \(R\) newtons.
  1. Find the value of \(R\). The lorry now travels up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The magnitude of the non-gravitational resistance to motion is \(R\) newtons. The lorry travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the new rate of working of the lorry's engine.
Edexcel M2 2013 January Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-05_876_757_125_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A ladder, of length 5 m and mass 18 kg , has one end \(A\) resting on rough horizontal ground and its other end \(B\) resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 1. The coefficient of friction between the ladder and the ground is \(\mu\). A woman of mass 60 kg stands on the ladder at the point \(C\), where \(A C = 3 \mathrm {~m}\). The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of \(\mu\).
Edexcel M2 2013 January Q4
10 marks Moderate -0.3
4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).
  1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
  2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
  3. Find
    1. the value of \(c\),
    2. the value of \(d\).
Edexcel M2 2013 January Q5
11 marks Standard +0.3
5. The point \(A\) lies on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 24 } { 25 }\). A particle \(P\) is projected from \(A\), up a line of greatest slope of the plane, with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg and the coefficient of friction between \(P\) and the plane is \(\frac { 5 } { 12 }\). The particle comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 1.5 \mathrm {~m}\). It then moves back down the plane to \(A\).
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\).
  3. Find the speed of \(P\) when it returns to \(A\).
Edexcel M2 2013 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-11_531_931_230_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ball is thrown from a point \(O\), which is 6 m above horizontal ground. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through \(O\), as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.
  1. Show that \(\tan \theta = 2.2\)
  2. Find the value of \(u\). The ball hits the ground \(T\) seconds after projection.
  3. Find the value of \(T\). Immediately before the ball hits the ground the direction of motion of the ball makes an angle \(\alpha\) with the horizontal.
  4. Find \(\alpha\).