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Edexcel M2 2013 January Q7
16 marks Challenging +1.2
7. A particle \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal floor when it collides directly with another particle \(B\), of mass \(3 m\), which is at rest on the floor. The coefficient of restitution between the particles is \(e\). The direction of motion of \(A\) is reversed by the collision.
  1. Find, in terms of \(e\) and \(u\),
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After being struck by \(A\) the particle \(B\) collides directly with another particle \(C\), of mass \(4 m\), which is at rest on the floor. The coefficient of restitution between \(B\) and \(C\) is \(2 e\). Given that the direction of motion of \(B\) is reversed by this collision,
  2. find the range of possible values of \(e\),
  3. determine whether there will be a second collision between \(A\) and \(B\).
Edexcel M2 2014 January Q2
5 marks Easy -1.3
2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-05_104_97_2613_1784}
Edexcel M2 2014 January Q4
4 marks Easy -1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-08_835_777_118_596} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at \(( 0,3 )\) and has a minimum at \(P ( 4,2 )\). On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( x + 4 )\),
  2. \(y = 2 \mathrm { f } ( x )\). On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.
Edexcel M2 2014 January Q6
11 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-12_650_885_255_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 7\) The line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.
    1. State the gradient of \(l _ { 1 }\)
    2. Write down the coordinates of the point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B ( 1,5 )\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle A B C = 90 ^ { \circ }\),
  2. find an equation of \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The rectangle \(A B C D\), shown shaded in Figure 2, has vertices at the points \(A , B , C\) and \(D\).
  3. Find the exact area of rectangle \(A B C D\).
Edexcel M2 2001 June Q1
5 marks Moderate -0.8
  1. At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where
$$\mathbf { r } = \left( t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t - 2 t ^ { 2 } \right) \mathbf { j } .$$ Show that the acceleration of \(P\) is constant and find its magnitude. \section*{2.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{7670376b-a7b6-4f8c-83b5-bad3f5ff5162-2_861_588_837_807}
\end{figure} Figure 1 shows a decoration which is made by cutting 2 circular discs from a sheet of uniform card. The discs are joined so that they touch at a point \(D\) on the circumference of both discs. The discs are coplanar and have centres \(A\) and \(B\) with radii 10 cm and 20 cm respectively.
  1. Find the distance of the centre of mass of the decoration from B. The point \(C\) lies on the circumference of the smaller disc and \(\angle C A B\) is a right angle. The decoration is freely suspended from C and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle between AB and the vertical.
Edexcel M2 2001 June Q3
9 marks Standard +0.8
3. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5 . The other end \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of \(30 ^ { \circ }\) with the wall. A man of mass \(5 m\) stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(k a\). Find the value of \(k\).
Edexcel M2 2001 June Q4
10 marks Moderate -0.8
4. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) vertical. A ball of mass 0.1 kg is hit by a bat which gives it an impulse of ( \(3.5 \mathbf { i } + 3 \mathbf { j }\) ) Ns. The velocity of the ball immediately after being hit is \(( 10 \mathbf { i } + 25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of the ball immediately before it is hit. In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.
  2. Find the greatest height of the ball above the ground in the subsequent motion. The ball is caught when it is again 1 m above the ground.
  3. Find the distance from the point where the ball is hit to the point where it is caught.
Edexcel M2 2001 June Q5
10 marks Standard +0.3
5. A child is playing with a small model of a fire-engine of mass 0.5 kg and a straight, rigid plank. The plank is inclined at an angle \(\alpha\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons.
When \(\alpha = 20 ^ { \circ }\) the fire-engine is projected with an initial speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and first comes to rest after travelling 2 m .
  1. Find, to 3 significant figures, the value of \(R\). When \(\alpha = 40 ^ { \circ }\) the fire-engine is again projected with an initial speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find how far the fire-engine travels before first coming to rest.
Edexcel M2 2001 June Q6
16 marks Standard +0.3
6. A particle \(A\) of mass \(2 m\) is moving with speed \(2 u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4 m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 3 } { 2 } u\).
  2. Find the speed of \(A\) after the collision. Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,
  3. find the range of possible values for \(e\).
    (8)
Edexcel M2 2001 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{7670376b-a7b6-4f8c-83b5-bad3f5ff5162-5_501_1284_425_386}
\end{figure} At time \(t = 0\) a small package is projected from a point \(B\) which is 2.4 m above a point \(A\) on horizontal ground. The package is projected with speed \(23.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). The package strikes the ground at the point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.
  1. Find the time taken for the package to reach \(C\). A lorry moves along the line \(A C\), approaching \(A\) with constant speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest \(T\) seconds later. The acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the lorry at time \(t\) seconds is given by $$a = - \frac { 1 } { 4 } t ^ { 2 } , \quad 0 \leq t \leq T .$$
  2. Find the speed of the lorry at time \(t\) seconds.
  3. Hence show that \(T = 6\).
  4. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry.
Edexcel M2 2002 June Q1
8 marks Moderate -0.8
  1. The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds is given by
$$\mathbf { v } = ( 3 t - 2 ) \mathbf { i } - 5 t \mathbf { j } .$$
  1. Show that the acceleration of \(P\) is constant.
    (2) At \(t = 0\), the position vector of \(P\) relative to a fixed origin O is \(3 \mathbf { i } \mathrm {~m}\).
  2. Find the distance of \(P\) from O when \(t = 2\).
Edexcel M2 2002 June Q2
8 marks Standard +0.3
2. A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) is given by $$a = \begin{cases} 4 t - t ^ { 2 } , & 0 \leq t \leq 3 , \\ \frac { 27 } { t ^ { 2 } } , & t > 3 . \end{cases}$$ At \(t = 0 , P\) is at rest. Find the speed of \(P\) when
  1. \(t = 3\),
  2. \(t = 6\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a89db1ee-073c-43a3-8480-44970e51c6e2-3_329_1198_391_515}
    \end{figure} Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m . When she reaches \(B\), her speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m . By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N ,
  3. find the work done by the cyclist in moving from \(A\) to \(B\). At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
  4. find the power generated by the cyclist at \(B\).
Edexcel M2 2002 June Q4
11 marks Standard +0.3
4. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{a89db1ee-073c-43a3-8480-44970e51c6e2-4_671_781_442_691}
A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(A B C D\), of side 10 cm , and removing the semi-circle with diameter \(A B\) from the square, as shown in Fig. 2.
  1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(A B\).
    [0pt] [The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac { 4 a } { 3 \pi }\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.
  2. Find, in degrees to one decimal place, the angle between CD and the vertical.
Edexcel M2 2002 June Q5
12 marks Moderate -0.3
5. A particle is projected from a point with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).
  1. Show that $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right) .$$ A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of \(45 ^ { \circ }\) with a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the shot as a particle moving freely under gravity,
  2. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground,
  3. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground.
Edexcel M2 2002 June Q6
13 marks Standard +0.3
6. A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(u\) when it collides directly with another small smooth ball \(B\) of mass \(3 m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.
  1. Find, in terms of \(e\) and \(u\). the speeds of \(A\) and \(B\) immediately after the collision. In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). Given that there is a second collision between \(A\) and \(B\),
  2. find the range of values of \(e\) for which the motion described is possible.
    (6) \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a89db1ee-073c-43a3-8480-44970e51c6e2-6_345_1133_440_524}
    \end{figure} A straight \(\log A B\) has weight \(W\) and length \(2 a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 5 } { 12 }\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6 . The log is modelled as a uniform rod and the cable as light.
  3. Show that the normal reaction on the \(\log\) at \(A\) is \(\frac { 2 } { 5 } W\).
  4. Find the value of \(\beta\). The tension in the cable is \(k W\).
  5. Find the value of \(k\).
Edexcel M2 2003 June Q1
5 marks Moderate -0.3
  1. 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 \(v = 6 t - 2 t ^ { 2 }\). When \(t = 0 , P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest after leaving \(O\).
  2. A tennis ball of mass 0.2 kg is moving with velocity \(( - 10 \mathbf { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a tennis racket. Immediately after being struck, the ball has velocity \(( 15 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
    1. the magnitude of the impulse exerted by the racket on the ball,
    2. the angle, to the nearest degree, between the vector \(\mathbf { i }\) and the impulse exerted by the racket,
    3. the kinetic energy gained by the ball as a result of being struck.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b6c32af8-4b73-4ac0-94a5-e046bda06939-2_473_721_1302_625}
    \end{figure} A uniform lamina \(A B C D\) is made by taking a uniform sheet of metal in the form of a rectangle \(A B E D\), with \(A B = 3 a\) and \(A D = 2 a\), and removing the triangle \(B C E\), where \(C\) lies on \(D E\) and \(C E = a\), as shown in Fig. 1.
  3. Find the distance of the centre of mass of the lamina from \(A D\). The lamina has mass \(M\). A particle of mass \(m\) is attached to the lamina at \(B\). When the loaded lamina is freely suspended from the mid-point of \(A B\), it hangs in equilibrium with \(A B\) horizontal.
  4. Find \(m\) in terms of \(M\).
    (4) \section*{4.} \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{b6c32af8-4b73-4ac0-94a5-e046bda06939-3_471_618_402_678}
    A uniform steel girder \(A B\), of mass 40 kg and length 3 m , is freely hinged at \(A\) to a vertical wall. The girder is supported in a horizontal position by a steel cable attached to the girder at \(B\). The other end of the cable is attached to the point \(C\) vertically above \(A\) on the wall, with \(\angle A B C = \alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). A load of mass 60 kg is suspended by another cable from the girder at the point \(D\), where \(A D = 2 \mathrm {~m}\), as shown in Fig. 2. The girder remains horizontal and in equilibrium. The girder is modelled as a rod, and the cables as light inextensible strings.
  5. Show that the tension in the cable \(B C\) is 980 N .
  6. Find the magnitude of the reaction on the girder at \(A\).
  7. Explain how you have used the modelling assumption that the cable at \(D\) is light.
Edexcel M2 2003 June Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b6c32af8-4b73-4ac0-94a5-e046bda06939-4_698_967_484_502}
\end{figure} A ball is thrown from a point 4 m above horizontal ground. The ball is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The ball hits the ground at a point which is a horizontal distance 8 m from its point of projection, as shown in Fig. 3. The initial speed of the ball is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the time of flight is \(T\) seconds.
  1. Prove that \(u T = 10\).
  2. Find the value of \(u\). As the ball hits the ground, its direction of motion makes an angle \(\phi\) with the horizontal.
  3. Find \(\tan \phi\).
    (5)
Edexcel M2 2003 June Q6
14 marks Standard +0.3
6. A girl and her bicycle have a combined mass of 64 kg . She cycles up a straight stretch of road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). She cycles at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When she is cycling at this speed, the resistance to motion from non-gravitational forces has magnitude 20 N .
  1. Find the rate at which the cyclist is working.
    (4) She now turns round and comes down the same road. Her initial speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the resistance to motion is modelled as remaining constant with magnitude 20 N . She free-wheels down the road for a distance of 80 m . Using this model,
  2. find the speed of the cyclist when she has travelled a distance of 80 m . The cyclist again moves down the same road, but this time she pedals down the road. The resistance is now modelled as having magnitude proportional to the speed of the cyclist. Her initial speed is again \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the resistance to motion has magnitude 20 N .
  3. Find the magnitude of the resistance to motion when the speed of the cyclist is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist works at a constant rate of 200 W.
  4. Find the magnitude of her acceleration when her speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (4)
Edexcel M2 2003 June Q7
15 marks Standard +0.3
7. A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2 m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.
  1. Find the speeds of \(A\) and \(B\) immediately after the collision.
  2. Find the range of possible values of \(e\). After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4 m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).
  3. Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\).
    (6) \section*{END}
Edexcel M2 2004 June Q1
7 marks Moderate -0.3
  1. A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion of the lorry has magnitude 750 N and the lorry's engine is working at a rate of 36 kW .
    1. Find the acceleration of the lorry when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry comes to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The magnitude of the resistance to motion from non-gravitational forces remains 750 N . The lorry moves up the hill at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate at which the lorry's engine is now working.
    (3)
Edexcel M2 2004 June Q2
9 marks Moderate -0.8
2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] A ball has mass 0.2 kg . It is moving with velocity ( 30 i ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. The bat exerts an impulse of \(( - 4 \mathbf { i } + 4 \mathbf { j } )\) Ns on the ball. Find
  1. the velocity of the ball immediately after the impact,
  2. the angle through which the ball is deflected as a result of the impact,
  3. the kinetic energy lost by the ball in the impact.
Edexcel M2 2004 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8e694174-b9a9-4018-8896-31a3b4f0d344-3_860_565_269_740}
\end{figure} Figure 1 shows a decoration which is made by cutting the shape of a simple tree from a sheet of uniform card. The decoration consists of a triangle \(A B C\) and a rectangle \(P Q R S\). The points \(P\) and \(S\) lie on \(B C\) and \(M\) is the mid-point of both \(B C\) and \(P S\). The triangle \(A B C\) is isosceles with \(A B = A C , B C = 4 \mathrm {~cm} , A M = 6 \mathrm {~cm} , P S = 2 \mathrm {~cm}\) and \(P Q = 3 \mathrm {~cm}\).
  1. Find the distance of the centre of mass of the decoration from \(B C\). The decoration is suspended from \(Q\) and hangs freely.
  2. Find, in degrees to one decimal place, the angle between \(P Q\) and the vertical.
Edexcel M2 2004 June Q4
10 marks Standard +0.3
4. At time \(t\) seconds, the velocity of a particle \(P\) is \([ ( 4 t - 7 ) \mathbf { i } - 5 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is at the point with position vector \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\).
  1. Find an expression for the position vector of \(P\) after \(t\) seconds, giving your answer in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { m }\). A second particle \(Q\) moves with constant velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( - 7 \mathrm { i } ) \mathrm { m }\).
  2. Prove that \(P\) and \(Q\) collide.
Edexcel M2 2004 June Q5
11 marks Standard +0.8
5. Two small smooth spheres, \(P\) and \(Q\), of equal radius, have masses \(2 m\) and \(3 m\) respectively. The sphere \(P\) is moving with speed \(5 u\) on a smooth horizontal table when it collides directly with \(Q\), which is at rest on the table. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(2 ( 1 + e ) u\). After the collision, \(Q\) hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f , 0 < f \leqslant 1\).
  2. Show that, when \(e = 0.4\), there is a second collision between \(P\) and \(Q\). Given that \(e = 0.8\) and there is a second collision between \(P\) and \(Q\),
  3. find the range of possible values of \(f\).
Edexcel M2 2004 June Q6
12 marks Challenging +1.2
6. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.6 . The other end \(B\) of the ladder rests against a smooth vertical wall. A builder of mass 10 m stands at the top of the ladder. To prevent the ladder from slipping, the builder's friend pushes the bottom of the ladder horizontally towards the wall with a force of magnitude \(P\). This force acts in a direction perpendicular to the wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 2 }\).
  1. Show that the reaction of the wall on the ladder has magnitude 7 mg .
  2. Find, in terms of \(m\) and \(g\), the range of values of \(P\) for which the ladder remains in equilibrium.