Past Exam Questions

A block of mass \(200\text{ kg}\) is pulled along rough horizontal ground at a constant speed of \(v\text{ m s}^{-1}\) by a horizontal rope attached to a winch. The coefficient of friction between the block and the ground is \(0.8\). The winch is working at a constant rate of \(4000\text{ W}\).

Find the value of \(v\).

A cyclist starts from rest and moves in a straight line with acceleration \(0.5\text{ m s}^{-2}\) for \(10\text{ s}\). They then accelerate at \(1.5\text{ m s}^{-2}\) for a distance of \(13\text{ m}\), reaching a speed of \(v\text{ m s}^{-1}\). They then travel at speed \(v\text{ m s}^{-1}\) for \(5\text{ s}\). They then travel a distance of \(28\text{ m}\) whilst decelerating uniformly to rest.

(a) Find the value of \(v\).

(b)(i) Find the total distance travelled by the cyclist.

(b)(ii) Find the average speed of the cyclist for the whole of their motion.

Coplanar forces of magnitudes \(F\text{ N}\), \(2F\text{ N}\) and \(20\text{ N}\) act at a point, as shown in the diagram.

Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).

A car of mass \(1600\text{ kg}\) passes through points \(A\) and \(B\) with speeds \(10\text{ m s}^{-1}\) and \(12\text{ m s}^{-1}\) respectively. The distance \(AB\) is \(2\text{ km}\). The heights of \(A\) and \(B\) above sea level are \(250\text{ m}\) and \(200\text{ m}\) respectively. The car’s engine does no work in moving from \(A\) to \(B\). There are two forces resisting the motion of the car, a braking force and an additional constant force of magnitude \(150\text{ N}\).

Use an energy method to find the work done by the braking force as the car moves from \(A\) to \(B\).

A particle \(P\) of mass \(0.3\text{ kg}\) is connected by a light inextensible string to a particle \(Q\) of mass \(0.2\text{ kg}\). The string joining the two particles is taut and is parallel to a line of greatest slope of a rough plane which is inclined at an angle of \(30^\circ\) to the horizontal. A constant force of magnitude \(12\text{ N}\) acts on \(Q\) and pulls the particles up the plane. The \(12\text{ N}\) force acts parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between each of the particles and the plane is \(0.4\).

(a) Find the magnitude of the acceleration of the particles and the tension in the string.

(b) At the instant when the speed of the particles is \(2\text{ m s}^{-1}\), the string breaks.

Find the time it takes from the instant the string breaks until \(P\) comes to instantaneous rest.

Two particles \(P\) and \(Q\) of masses \(4m\text{ kg}\) and \(3m\text{ kg}\) respectively are at rest on smooth horizontal ground. \(P\) is projected directly towards \(Q\) with speed \(6.5\text{ m s}^{-1}\). After the particles collide, they move in the same direction and the speeds of \(P\) and \(Q\) are \(2\text{ m s}^{-1}\) and \(v\text{ m s}^{-1}\) respectively.

(a)(i) Show that \(v=6\).

(a)(ii) Given that the energy lost in the collision between \(P\) and \(Q\) is \(45\text{ J}\), find the value of \(m\).

After \(Q\) has moved a distance of \(9\text{ m}\) from the point at which \(P\) and \(Q\) first collided, it hits a vertical wall perpendicular to the direction of motion of \(Q\) and rebounds. The speed of \(Q\) after hitting the wall is \(0.5\text{ m s}^{-1}\).

(b) Find the distance from the wall to the point at which the second collision between \(P\) and \(Q\) occurs.

A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\text{ s}\) is \(v\text{ m s}^{-1}\), where

\[ v=t-7t^{\frac12}+12. \]

(a) Find the acceleration of \(P\) at \(t=4\).

(b) Find the values of \(t\) at which \(P\) is instantaneously at rest.

(c) Find the total distance travelled by \(P\) in the interval \(0\leq t\leq25\).

A car of mass \(960\text{ kg}\) is moving on a straight horizontal road. There is a constant force of magnitude \(620\text{ N}\) resisting the motion of the car.

(a) Calculate the power developed by the engine of the car when it is moving at a constant speed of \(25\text{ m s}^{-1}\).

(b) Given that the power is suddenly increased by \(12\text{ kW}\), find the instantaneous acceleration of the car.

Two particles, \(A\) and \(B\), of masses \(2.5\text{ kg}\) and \(3.5\text{ kg}\) respectively are at rest on a straight smooth horizontal track. Particle \(B\) is situated \(9\text{ m}\) from a vertical wall which is fixed at right angles to the track.

Particle \(A\) is projected directly towards \(B\) with a speed of \(3\text{ m s}^{-1}\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(C\). Particle \(C\) then collides directly with the wall and rebounds. The collision of \(C\) with the wall reduces the speed of \(C\) by \(25\%\).

(a) Find the speed of \(C\) after it rebounds from the wall.

(b) Hence find the time from the instant at which \(A\) and \(B\) collide until \(C\) is once again a distance of \(9\text{ m}\) from the wall.

Two particles, \(A\) and \(B\), of masses \(0.5\text{ kg}\) and \(0.8\text{ kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the end of a rough horizontal plane and to the top of a smooth inclined plane. Particle \(A\) is held on the horizontal plane, while \(B\) lies on the inclined plane, which makes an angle of \(55^\circ\) with the horizontal. The string is in the same vertical plane as a line of greatest slope of the inclined plane (see diagram).

Particle \(A\) is released from rest with the string taut. Particle \(B\) moves \(0.3\text{ m}\) down the inclined plane in \(0.4\text{ s}\).

(a) Find the tension in the string.

(b) Find the coefficient of friction between \(A\) and the horizontal plane.

(c) After \(B\) has been moving for \(0.4\text{ s}\), the string suddenly breaks. Given that \(A\) subsequently comes to rest on the horizontal plane, find the work done by the frictional force in bringing \(A\) to rest.

You may assume that \(A\) does not reach the pulley.

A particle \(P\) is projected from a point \(A\) with speed \(4\text{ m s}^{-1}\) up a line of greatest slope of a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan\theta=\frac43\).

\(P\) comes to instantaneous rest at a point \(B\) on the plane (see diagram).

The coefficient of friction between \(P\) and the plane is \(\frac13\).

(a) Using an energy method, show that the distance \(AB\) is \(0.8\text{ m}\).

(b) After coming to instantaneous rest at \(B\), the particle slides down the plane.

Find the total time from the instant at which \(P\) is projected from \(A\) until it returns to \(A\).

The points \(A\) and \(B\) are at the same vertical height \(h\text{ m}\) above horizontal ground. A particle \(P\) is released from rest from \(A\). One second later, a particle \(Q\) is projected vertically downwards from \(B\) with speed \(18\text{ m s}^{-1}\).

Given that \(P\) and \(Q\) reach the ground at the same time, find the value of \(h\).

A crate of weight \(W\text{ N}\) is at rest on rough horizontal ground. The crate is pulled at a constant acceleration of \(1.5\text{ m s}^{-2}\) along the ground in a straight line by a light rope. The rope is inclined at an angle of \(\theta\) to the horizontal, where \(\theta=\sin^{-1}\frac5{13}\). The tension in the rope is \(26\text{ N}\) (see diagram). The coefficient of friction between the crate and the ground is \(0.25\).

(a) Find the value of \(W\).

(b) When the crate reaches a point \(A\), the rope is removed. The speed of the crate at \(A\) is \(8\text{ m s}^{-1}\).

The crate comes to rest at a point \(B\).

Find the distance \(AB\).

A particle \(P\) moves in a straight line. The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\), where \(t\geq0\), is given by

\[ v=k_1(4t+1)^{\frac12}-\frac14(2t+1)^2+k_2, \]

where \(k_1\) and \(k_2\) are constants. When \(t=1.25\) the deceleration of \(P\) is \(0.5\text{ m s}^{-2}\).

(a) Find the value of \(k_1\).

(b) Given that \(P\) is only at instantaneous rest when \(t=3.5\), find the distance travelled by \(P\) in the interval \(0\leq t\leq1\).

There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows.
\[\begin{array}{llllllll}
1.75 & 1.72 & 1.62 & 1.70 & 1.82 & 1.75 & 1.68 & 1.84
\end{array}\]

You may assume that heights of students are normally distributed.
(i) Test, at the \(5 \%\) significance level, whether the population mean height of students at this college is greater than 1.70 metres.

(ii) Find a 95\% confidence interval for the population mean height of students at this college.