Questions M2 (1537 questions)

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Edexcel M2 Q8
15 marks Standard +0.3
In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of 8 ms\(^{-1}\) in a direction making an angle of 30° with the horizontal. \includegraphics{figure_8}
  1. Find the time taken by the ball to travel 6 m horizontally. [2 marks]
  2. Showing your method clearly, decide whether or not the ball will hit the coconut. [4 marks]
  3. Find the greatest height reached by the ball above the level from which it was thrown. [4 marks]
  4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of 8 m s\(^{-1}\). [3 marks]
  5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut. [2 marks]
Edexcel M2 Q1
6 marks Moderate -0.3
A ball, of mass \(m\) kg, is moving with velocity \((5\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) when it receives an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) Ns. Immediately after the impulse is applied, the ball has velocity \((3\mathbf{i} + k\mathbf{j})\) ms\(^{-1}\). Find the values of the constants \(k\) and \(m\). [6 marks]
Edexcel M2 Q2
6 marks Moderate -0.3
A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \((12t - 15)\) ms\(^{-2}\). Find
  1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\), [3 marks]
  2. the value of \(t\) when the speed of \(P\) is 36 ms\(^{-1}\). [3 marks]
Edexcel M2 Q3
7 marks Standard +0.3
A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]
Edexcel M2 Q4
9 marks Moderate -0.8
A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf{r}\) m relative to \(O\) after \(t\) seconds is given by \(\mathbf{r} = at^2\mathbf{i} + bt\mathbf{j}\). 60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \((90\mathbf{i} + 30\mathbf{j})\) m.
  1. Find the values of the constants \(a\) and \(b\). [3 marks]
  2. Find the speed of \(P\) when it is at \(Q\). [4 marks]
  3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\). [2 marks]
Edexcel M2 Q5
10 marks Standard +0.3
A lorry of mass 4200 kg can develop a maximum power of 84 kW. On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at 20 ms\(^{-1}\) the resisting force has magnitude 2400 N. Find the maximum speed of the lorry when it is
  1. travelling on a horizontal road, [4 marks]
  2. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{7}\). [6 marks]
Edexcel M2 Q6
11 marks Standard +0.3
Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
  2. Show that \(e > \frac{1}{9}\). [2 marks]
\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).
  1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]
Edexcel M2 Q7
11 marks Standard +0.3
A uniform lamina is in the form of a trapezium \(ABCD\), as shown. \(AB\) and \(DC\) are perpendicular to \(BC\). \(AB = 17\) cm, \(BC = 21\) cm and \(CD = 8\) cm. \includegraphics{figure_7}
  1. Find the distances of the centre of mass of the lamina from
    1. \(AB\),
    2. \(BC\). [8 marks]
The lamina is freely suspended from \(C\) and rests in equilibrium.
  1. Find the angle between \(CD\) and the vertical. [3 marks]
Edexcel M2 Q8
15 marks Moderate -0.3
A stone, of mass 1.5 kg, is projected horizontally with speed 4 ms\(^{-1}\) from a height of 7 m above horizontal ground.
  1. Show that the stone travels about 4.78 m horizontally before it hits the ground. [4 marks]
  2. Find the height of the stone above the ground when it has travelled half of this horizontal distance. [4 marks]
  3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground. [2 marks]
  4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground. [3 marks]
  5. State two modelling assumptions that you have made in answering this question. [2 marks]
Edexcel M2 Q1
4 marks Moderate -0.3
A small ball \(A\) is moving with velocity \((7\mathbf{i} + 12\mathbf{j})\) ms\(^{-1}\). It collides in mid-air with another ball \(B\), of mass \(0.4\) kg, moving with velocity \((-\mathbf{i} + 7\mathbf{j})\) ms\(^{-1}\). Immediately after the collision, \(A\) has velocity \((-3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) and \(B\) has velocity \((6.5\mathbf{i} + 13\mathbf{j})\) ms\(^{-1}\). Calculate the mass of \(A\). [4 marks]
Edexcel M2 Q2
6 marks Standard +0.3
A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]
Edexcel M2 Q3
7 marks Standard +0.3
An engine of mass \(20\,000\) kg climbs a hill inclined at \(10°\) to the horizontal. The total non-gravitational resistance to its motion has magnitude \(35\,000\) N and the maximum speed of the engine on the hill is \(15\) ms\(^{-1}\).
  1. Find, in kW, the maximum rate at which the engine can work. [4 marks]
  2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. [3 marks]
Edexcel M2 Q4
7 marks Moderate -0.3
Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \((2t + 4)\mathbf{i} + (kt^2 - 5)\mathbf{j}\).
  1. Find the value of \(k\) if \(P\) takes \(4\) seconds to reach \(Y\). [3 marks]
  2. Show that \(P\) has constant acceleration and find the magnitude and direction of this acceleration. [4 marks]
Edexcel M2 Q5
10 marks Standard +0.8
Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]
Edexcel M2 Q6
11 marks Standard +0.8
A rectangular piece of cardboard \(ABCD\), measuring \(30\) cm by \(12\) cm, has a semicircle of radius \(5\) cm removed from it as shown. \includegraphics{figure_6}
  1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(AB\) and from \(BC\). [7 marks]
The remaining cardboard is suspended from \(A\) and hangs in equilibrium.
  1. Find the angle made by \(AB\) with the vertical. [4 marks]
Edexcel M2 Q7
15 marks Standard +0.3
A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v\) ms\(^{-1}\), is given at time \(t\) seconds after firing by the formula $$v = pt^2 + qt.$$ \(5\) seconds after firing, the rocket is travelling at \(500\) ms\(^{-1}\). \(30\) seconds after firing, the rocket is travelling at \(12\,000\) ms\(^{-1}\).
  1. Find the constants \(p\) and \(q\). [4 marks]
  2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\). [2 marks]
  3. Find the initial acceleration of the rocket. [2 marks]
  4. Find the distance of the rocket from \(O\) \(30\) seconds after firing. [4 marks]
From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12\,000\) ms\(^{-1}\).
  1. Find the average speed of the rocket during its first \(50\) seconds of motion. [3 marks]
Edexcel M2 Q8
15 marks Standard +0.8
A golf ball is hit with initial velocity \(u\) ms\(^{-1}\) at an angle of \(45°\) above the horizontal. The ball passes over a building which is \(15\) m tall at a distance of \(30\) m horizontally from the point where the ball was hit.
  1. Find the smallest possible value of \(u\). [7 marks]
When \(u\) has this minimum value,
  1. show that the ball does not rise higher than the top of the building. [4 marks]
  2. Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
  3. Briefly describe two modelling assumptions that you have made. [2 marks]
Edexcel M2 Q1
7 marks Moderate -0.3
The acceleration of a particle \(P\) is \((8t - 18)\) ms\(^{-2}\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving. At time \(t = 3\), \(P\) has speed \(2\) ms\(^{-1}\). Find
  1. the velocity of \(P\) at time \(t\), [4 marks]
  2. the values of \(t\) when \(P\) is instantaneously at rest. [3 marks]
Edexcel M2 Q2
7 marks Moderate -0.3
A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed 12 ms\(^{-1}\) through a pipe of radius 4 cm. Find
  1. the potential and kinetic energy given to the water each second, [5 marks]
  2. the rate, in kW, at which the pump is working. [2 marks]
[1 m\(^3\) of water has a mass of 1000 kg.]
Edexcel M2 Q3
7 marks Moderate -0.3
A particle \(P\) of mass 3 kg has position vector \(\mathbf{r} = (2t^2 - 4t)\mathbf{i} + (1 - t^2)\mathbf{j}\) m at time \(t\) seconds.
  1. Find the velocity vector of \(P\) when \(t = 3\). [3 marks]
  2. Find the magnitude of the force acting on \(P\), showing that this force is constant. [4 marks]
Edexcel M2 Q4
9 marks Standard +0.3
\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCDE\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm.
  1. Find the distance of the centre of mass of the lamina from \(ED\). [4 marks]
The lamina has mass \(m\). A particle \(P\) is attached to the lamina at \(B\). The lamina is then suspended freely from \(A\) and hangs in equilibrium with \(AD\) vertical.
  1. Find, in terms of \(m\), the mass of \(P\). [5 marks]
Edexcel M2 Q5
14 marks Standard +0.3
A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.
  1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
  2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
  3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]
When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,
  1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
  2. Show that tension in the tow-bar remains unchanged. [3 marks]
Edexcel M2 Q6
15 marks Standard +0.3
Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.
  1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
  2. Find the time for which the ball is in flight. [4 marks]
  3. Find the horizontal distance travelled by the ball. [3 marks]
  4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
  5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]
Edexcel M2 Q7
16 marks Standard +0.8
Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.
  1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
  2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]
Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).
  1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]
Edexcel M2 Q1
5 marks Moderate -0.8
A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]