Questions — Edexcel M3 (469 questions)

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Edexcel M3 Q4
4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).
  1. For the motion of the particle write down
    1. the period,
    2. the amplitude.
  2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
  3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
    (4 marks)
Edexcel M3 Q5
5. When a particle of mass \(M\) is at a distance of \(x\) metres from the centre of the moon, the gravitational force, \(F\) N, acting on it and directed towards the centre of the moon is given by $$F = \frac { \left( 4.90 \times 10 ^ { 12 } \right) M } { x ^ { 2 } }$$ A rocket is projected vertically into space from a point on the surface of the moon with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the radius of the moon is \(\left( 1.74 \times 10 ^ { 6 } \right) \mathrm { m }\),
  1. show that the speed of the rocket, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is \(x\) metres from the centre of the moon is given by $$v ^ { 2 } = u ^ { 2 } + \frac { a } { x } - b$$ where \(a\) and \(b\) are constants which should be found correct to 3 significant figures.
  2. Find, correct to 2 significant figures, the minimum value of \(u\) needed for the rocket to escape the moon's gravitational attraction.
Edexcel M3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_526_620_196_598} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.
  1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
    (7 marks)
    The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
  2. Find tan \(\alpha\).
    (6 marks)
Edexcel M3 Q7
7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,
  1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
  2. show that \(\mu = 0.227\), correct to 3 significant figures,
  3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END
Edexcel M3 Q1
  1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the toy in terms of \(t\).
  2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
  3. Explain why this model is unlikely to be realistic for large values of \(t\).
Edexcel M3 Q2
2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).
  1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
  2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).
Edexcel M3 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-2_382_796_1640_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.
  1. Find the tension in the string.
  2. Find the length of the string.
Edexcel M3 Q4
4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
  1. Show that the period of the motion is \(\pi\) seconds.
  2. Find the amplitude of the motion.
  3. Hence, find the greatest speed of the particle.
Edexcel M3 Q5
5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-3_307_1072_993_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.
  1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
  2. Find the mass of the object.
  3. Find the elastic potential energy of the string when the object is suspended from it.
Edexcel M3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
  2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
    (3 marks)
Edexcel M3 Q7
7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).
  1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
  2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
  3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.
Edexcel M3 Q1
  1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
    1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
    Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
  2. find the value of \(T\) correct to 3 significant figures.
Edexcel M3 Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.
  1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
  2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.
Edexcel M3 Q3
3. A car starts from rest at the point \(O\) and moves along a straight line. The car accelerates to a maximum velocity, \(V \mathrm {~ms} ^ { - 1 }\), before decelerating and coming to rest again at the point \(A\). The acceleration of the car during this journey, \(a \mathrm {~ms} ^ { - 2 }\), is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where \(x\) is the distance in metres of the car from \(O\).
Using this model and given that the car is travelling at \(16 \mathrm {~ms} ^ { - 1 }\) when it is 40 m from \(O\),
  1. find \(k\),
  2. show that \(V = 41\), correct to 2 significant figures,
  3. find the distance \(O A\).
Edexcel M3 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\).
\(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),
  1. show that the work done against friction is 10.78 J ,
  2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
  3. find \(\lambda\).
Edexcel M3 Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.
  1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
  2. find the value of \(k\).
  3. Explain why it may be advantageous to make the base of the flask from a more dense material.
    (2 marks)
Edexcel M3 Q6
6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(A B = 7.5 \mathrm {~m}\).
  2. Find the period of the motion.
  3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
  4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.
Edexcel M3 2003 January Q3
  1. Show that the distance \(d\) of the centre of mass of the toy from its lowest point \(O\) is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
  2. Find \(h\) in terms of \(r\).
    (3)
Edexcel M3 Q2
  1. Show that \(\lambda = 4 m g \sin \alpha\). The particle is now moved and held at rest at \(A\) with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.
  2. Find the greatest distance from \(A\) that the particle reaches down the plane.