Edexcel M2 (Mechanics 2)

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

A particle of mass 4 kg is moving in a straight horizontal line. There is a constant resistive force of magnitude \(R\) newtons. The speed of the particle is reduced from 25 m s\(^{-1}\) to rest over a distance of 200 m. Use the work-energy principle to calculate the value of \(R\). [4]

The velocity v m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

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 3i m.

1. Find the distance of \(P\) from \(O\) when \(t = 2\). [6]

Three particles of mass \(3m\), \(5m\) and \(2m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).

1. Show that \(λ = 2\). [4]

1. Calculate the value of \(k\). [3]

\includegraphics{figure_1} 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 \(B\). [5]

The point \(C\) lies on the circumference of the smaller disc and \(\angle CAB\) is a right angle. The decoration is freely suspended from \(C\) and hangs in equilibrium.

1. Find, in degrees to one decimal place, the angle between \(AB\) and the vertical. [4]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]

1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when

1. \(t = 3\), [3]

1. \(t = 6\). [5]

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 m 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 the point \(B\), her speed is 5 m 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,

1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]

At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),

1. find the power generated by the cyclist at \(B\). [4]

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(j\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(j\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]

1. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(α\) with the horizontal, where \(\tan α = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{4}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(0.15\) and \(B\) 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 \(30°\) with the wall. A man of mass \(5m\) 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 \(6a\). Find the value of \(k\). [9]

A particle \(P\) of mass \(0.3\) kg is moving under the action of a single force \(F\) newtons. At time \(t\) seconds the velocity of \(P\), v m s\(^{-1}\), is given by $$\mathbf{v} = 3t^2\mathbf{i} + (6t - 4)\mathbf{j}.$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]

When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).

1. Find the position vector of \(B\). [5]

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 \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).

1. Find the velocity of the ball immediately before it is hit. [3]

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.

1. Find the greatest height of the ball above the ground in the subsequent motion. [3]

The ball is caught when it is again 1 m above the ground.

1. Find the distance from the point where the ball is hit to the point where it is caught. [4]

\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The midpoint of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).

1. Show that \(NG = \frac{8}{7}a\). [7]

The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,

1. calculate the value of \(k\). [4]

\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) 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 \(AB\). [7]

[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3π}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.

1. Find, in degrees to one decimal place, the angle between \(CD\) and the vertical. [4]

\includegraphics{figure_2} A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(v\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.

1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{3}{4}u\) and \(\frac{5}{4}u\) respectively. [7]

After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.

1. Find the value of \(e\). [7]

1. Explain why there must be a third collision between \(P\) and \(Q\). [1]

Show that \(GX = \frac{44}{63}a\). [6] The mass of the lamina is \(M\). A particle of mass \(λM\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

1. Find the value of \(λ\). [3]

TURN OVER FOR QUESTION 7

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 \(α\) 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 \(α = 20°\) the fire-engine is projected with an initial speed of \(5\) m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(α = 40°\) the fire-engine is again projected with an initial speed of \(5\) m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(θ\) to the horizontal, where \(\tan θ = \frac{5}{3}\). The pole is modelled as a uniform rod.

1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]

The vertical component of the force exerted by the wall on the pole is \(\frac{1}{2}W\). Find

1. x in terms of \(a\), [3]

1. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]

A particle is projected from a point with speed \(u\) at an angle of elevation \(α\) 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 $$y = x \tan α - \frac{gx^2}{2u^2}(1 + \tan^2 α).$$ [5]

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 \(45°\) with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]

1. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(v\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) 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. [7]

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 \(\frac{1}{2}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find

1. \(v\) in terms of \(t\), [4]

1. the distance between the two points where \(P\) is instantaneously at rest. [7]

A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) 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 \(\frac{3}{2}u\). [6]

1. Find the speed of \(A\) after the collision. [2]

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,

1. find the range of possible values for \(e\). [8]

A smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,

1. show that the speed of \(Q\) immediately after the collision is \(\frac{1}{3}(1 + e)u\), [5]

1. find the range of possible values of \(e\). [4]

Given that \(e = \frac{1}{4}\),

1. find the loss of kinetic energy in the collision. [4]

1. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]

TURN OVER FOR QUESTION 7

\includegraphics{figure_2} 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 \(23.75\) m s\(^{-1}\) at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{5}\). The package strikes the ground at 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 \(C\). [5]

A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest 7 seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq 7.$$

1. Find the speed of the lorry at time \(t\) seconds. [3]

1. Hence show that \(T = 6\). [3]

1. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]

END

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of \(60°\), as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]

1. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]

1. find the distance \(OC\). [6]

END

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). 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 \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). The cable makes an angle \(β\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is \(\frac{1}{3}\). The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{3}{4}W\). [6]

1. Find the value of \(β\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

END