Questions — Edexcel (10514 questions)

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Edexcel M3 Q3
9 marks Challenging +1.2
Two particles \(A\) and \(B\), of masses \(M\) kg and \(m\) kg respectively, are connected by a light inextensible string passing over a smooth fixed pulley. \(B\) is placed on a smooth horizontal table and \(A\) hangs freely, as shown. \(B\) is attached to a spring of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed to a vertical wall. \includegraphics{figure_3} The system starts to move from rest when the string is taut and the spring neither extended nor compressed. \(A\) does not reach the ground, nor does \(B\) reach the pulley, during the motion.
  1. Show that the maximum extension of the spring is \(\frac{2Mgl}{\lambda}\) m. [3 marks]
  2. If \(M = 3\), \(m = 1.5\) and \(\lambda = 35l\), find the speed of \(A\) when the extension in the spring is \(0.5\) m. [6 marks]
Edexcel M3 Q4
11 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg moves along a straight line under the action of a force of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(OP = x\) m. \(P\) starts from rest at \(A\), at a distance \(a\) m from \(O\). When \(OP = x\) m, the speed of \(P\) is \(v\) ms\(^{-1}\).
  1. Show that \(v = \sqrt{\frac{2k(a-x)}{ax}}\). [6 marks]
\(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac{1}{2}\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds Assuming the result that, for \(0 \leq x \leq 1\), \(\int \sqrt{\frac{x}{1-x}} dx = \arcsin(\sqrt{x}) - \sqrt{x(1-x^2)} + \text{constant}\),
  1. find the value of \(T\). [5 marks]
Edexcel M3 Q5
13 marks Standard +0.8
A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.
  1. Show that when its speed is 8.28 ms\(^{-1}\), there is no sideways force acting on the car. [4 marks]
  2. When the speed of the car is 12.5 ms\(^{-1}\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]
Edexcel M3 Q6
13 marks Standard +0.8
The diagram shows a particle \(P\) of mass \(m\) kg moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r\) m which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h\) m below the top of the bowl. \includegraphics{figure_6}
  1. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac{mgr}{h}\) N. [4 marks]
  2. Find, in terms of \(g\), \(h\) and \(r\), the constant speed of \(P\). [4 marks]
The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(mg\) applied tangentially as shown. The reaction of the surface of the sphere on \(P\) now has magnitude \(S\) N. \includegraphics{figure_6b}
  1. Given that \(r = 2h\), prove that \(S < \frac{1}{6}R\). [5 marks]
Edexcel M3 Q7
15 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.
  1. Find the coefficient of friction between \(P\) and the table. [2 marks]
\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
  1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]
If \(P\) is held at the point where the extension in the string is \(l\) m and then released,
  1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
  2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]
Edexcel M3 Q1
6 marks Moderate -0.3
A cyclist travels on a banked track inclined at \(8°\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v\) ms\(^{-1}\). If there is no sideways frictional force on the cycle, calculate the value of \(v\). [6 marks]
Edexcel M3 Q2
9 marks Standard +0.3
The figure shows a particle \(P\), of mass 0·8 kg, attached to the ends of two light elastic strings. \(AP\) has natural length 20 cm and modulus of elasticity \(\lambda\) N. \(BP\) has natural length 20 cm and modulus of elasticity \(\mu\) N. \(A\) and \(B\) are fixed to points on the same horizontal level so that \(AB = 50\) cm. When \(P\) is suspended in equilibrium, \(AP = 30\) cm and \(BP = 40\) cm. Calculate the values of \(\lambda\) and \(\mu\). \includegraphics{figure_2} [9 marks]
Edexcel M3 Q3
10 marks Challenging +1.8
Suraiya, whose mass is \(m\) kg, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed 0·2 ms\(^{-1}\). She continues to move in the same straight line, the only force acting on her being a resistance of magnitude \(mv^2 \sin \left(\frac{t}{100}\right)\) N, where \(v\) ms\(^{-1}\) is her speed at time \(t\) seconds after entering the pool and \(0 \leq t \leq 50\pi\).
  1. Find an expression for \(v\) in terms of \(t\). [7 marks]
  2. Calculate her greatest and least speeds during her motion. [3 marks]
Edexcel M3 Q4
12 marks Challenging +1.2
A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown. \includegraphics{figure_4}
  1. Show by integration that the centre of mass of the lamina is at a distance \(\frac{\pi^2 - 4}{2\pi}\) from the \(y\)-axis. [9 marks]
Given that the centre of mass is at a distance 0·75 units from the \(x\)-axis, and that \(P\) is the point \((0, 2)\) and \(O\) is the origin \((0, 0)\),
  1. find, to the nearest degree, the angle between the line \(OP\) and the vertical when the lamina is freely suspended from \(P\). [3 marks]
Edexcel M3 Q5
12 marks Standard +0.8
A particle \(P\), of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is 0·5. \(P\) is connected to a particle \(Q\), of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point \(O\) on the table, such that the distance \(OQ\) is 0·4 m. \(Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium. \includegraphics{figure_5}
  1. Calculate the angle \(\theta\) which \(OQ\) makes with the vertical. [4 marks]
  2. Show that the speed of \(Q\) is 1·33 ms\(^{-1}\). [3 marks]
The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed 0·84 ms\(^{-1}\), at a constant distance \(r\) m from \(O\) but tending to slip away from \(O\).
  1. Find the value of \(r\). [5 marks]
Edexcel M3 Q6
12 marks Standard +0.3
The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. \includegraphics{figure_6} Each spring has natural length \(l\) m and modulus of elasticity \(\lambda\) N.
  1. Show that \(\lambda = 392l\). [2 marks]
The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.
  1. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\). [5 marks]
  2. Calculate the number of oscillations made per second in this motion. [3 marks]
  3. Find the maximum acceleration which the mass experiences during the motion. [2 marks]
Edexcel M4 2002 January Q1
4 marks Moderate -0.8
A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream. Find
  1. the speed of the river, [2]
  2. the speed of the swimmer relative to the water. [2]
Edexcel M4 2002 January Q2
8 marks Challenging +1.2
A ball of mass \(m\) is thrown vertically upwards from the ground with an initial speed \(u\). When the speed of the ball is \(v\), the magnitude of the air resistance is \(mkv\), where \(k\) is a positive constant. By modelling the ball as a particle, find, in terms of \(u\), \(k\) and \(g\), the time taken for the ball to reach its greatest height. [8]
Edexcel M4 2002 January Q3
10 marks Challenging +1.2
A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta\), \(\theta < 45°\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.
  1. Show that \(e = \tan^2 \theta\). [6]
  2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(mu \sec \theta\). [4]
Edexcel M4 2002 January Q4
11 marks Challenging +1.2
A pilot flying an aircraft at a constant speed of 2000 kmh\(^{-1}\) detects an enemy aircraft 100 km away on a bearing of 045°. The enemy aircraft is flying at a constant velocity of 1500 kmh\(^{-1}\) due west. Find
  1. the course, as a bearing to the nearest degree, that the pilot should set up in order to intercept the enemy aircraft,
  2. the time, to the nearest s, that the pilot will take to reach the enemy aircraft. [11]
Edexcel M4 2002 January Q5
12 marks Challenging +1.2
\includegraphics{figure_1} A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal table. The sphere \(S\) collides with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(km\), \(k > 1\), which is at rest on the table. The coefficient of restitution between the spheres is \(e\). Immediately before the spheres collide the direction of motion of \(S\) makes an angle \(\theta\) with the line joining their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of \(S\) and \(T\) are perpendicular.
  1. Show that \(e = \frac{1}{k}\). [6]
Given that \(k = 2\) and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,
  1. find the value of \(\theta\). [6]
Edexcel M4 2002 January Q6
15 marks Standard +0.8
\includegraphics{figure_2} In a simple model of a shock absorber, a particle \(P\) of mass \(m\) kg is attached to one end of a light elastic horizontal spring. The other end of the spring is fixed at \(A\) and the motion of \(P\) takes place along a fixed horizontal line through \(A\). The spring has natural length \(L\) metres and modulus of elasticity \(2mL\) newtons. The whole system is immersed in a fluid which exerts a resistance on \(P\) of magnitude \(3mv\) newtons, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) seconds. The compression of the spring at time \(t\) seconds is \(x\) metres, as shown in Fig. 2.
  1. Show that $$\frac{\text{d}^2 x}{\text{d}t^2} + 3\frac{\text{d}x}{\text{d}t} + 2x = 0.$$ [4]
Given that when \(t = 0\), \(x = 2\) and \(\frac{\text{d}x}{\text{d}t} = -4\),
  1. find \(x\) in terms of \(t\). [8]
  2. Sketch the graph of \(x\) against \(t\). [2]
  3. State, with a reason, whether the model is realistic. [1]
Edexcel M4 2002 January Q7
15 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\), of mass \(m\) and length \(2a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(AC = 4a\). A light elastic string, of natural length \(2a\) and modulus of elasticity \(\frac{1}{4}mg\), joins \(B\) to \(C\). The rod \(AB\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$-mga[\cos \theta + \sqrt{(5 - 4 \cos \theta)}] + \text{constant}.$$ [9]
  2. Hence determine the values of \(\theta\) for which the system is in equilibrium. [6]
Edexcel M4 2003 January Q1
6 marks Standard +0.8
A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of 0.75 m s\(^{-1}\). The boy can walk at a maximum speed of 1 m s\(^{-1}\). Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this. [6]
Edexcel M4 2003 January Q2
7 marks Standard +0.8
Boat \(A\) is sailing due east at a constant speed of 10 km h\(^{-1}\). To an observer on \(A\), the wind appears to be blowing from due south. A second boat \(B\) is sailing due north at a constant speed of 14 km h\(^{-1}\). To an observer on \(B\), the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction. [7]
Edexcel M4 2003 January Q3
11 marks Challenging +1.2
A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(mkv^2\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. [11]
Edexcel M4 2003 January Q4
16 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a uniform rod \(AB\), of mass \(m\) and length \(4a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{1}{4}mg\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle BAC = 2\theta\), where \(\theta < \frac{\pi}{4}\). Given that \(AC\) is always horizontal,
  1. show that the potential energy of the system is $$\frac{mga}{8}(16\sin 2\theta + 3\cot^2 \theta - 6\cot \theta) + \text{constant}.$$ [7]
  2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\). [6]
  3. Determine whether this position of equilibrium is stable or unstable. [3]
Edexcel M4 2003 January Q5
17 marks Challenging +1.3
A particle \(P\) moves in a straight line. At time \(t\) seconds its displacement from a fixed point \(O\) on the line is \(x\) metres. The motion of \(P\) is modelled by the differential equation $$\frac{\text{d}^2 x}{\text{d}t^2} + 2\frac{\text{d}x}{\text{d}t} + 2x = 12\cos 2t - 6\sin 2t.$$ When \(t = 0\), \(P\) is at rest at \(O\).
  1. Find, in terms of \(t\), the displacement of \(P\) from \(O\). [11]
  2. Show that \(P\) comes to instantaneous rest when \(t = \frac{\pi}{4}\). [2]
  3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac{\pi}{4}\). [2]
  4. Find the approximate period of the motion for large values of \(t\). [2]
Edexcel M4 2003 January Q6
18 marks Challenging +1.8
\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).
  1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
  2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
  3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]
For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,
  1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
  2. find the ratio \(d_1 : d_2\). [5]
Edexcel M4 2004 January Q1
5 marks Standard +0.3
A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v\) m s\(^{-1}\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2v\) N. Find the distance moved by \(P\) while slowing down from 5 m s\(^{-1}\) to 2 m s\(^{-1}\). [5]