Past Exam Questions

A particle A moves in a straight line with constant speed 10 m s^-1. Two seconds after A passes a point O on the line, a particle B passes through O, moving along the line in the same direction as A. Particle B has speed 16 m s^-1 at O and has a constant deceleration of 2 m s^-2.

(i) Find expressions, in terms of t, for the displacement from O of each particle t s after B passes through O.

(ii) Find the distance between the particles when B comes to instantaneous rest.

(iii) Find the minimum distance between the particles.

A particle P starts from a fixed point O and moves in a straight line. At time t s after leaving O, the velocity v m s^-1 of P is given by v = 6t - 0.3t². The particle comes to instantaneous rest at point X.

1. Find the distance OX.

A second particle Q starts from rest from O, at the same instant as P, and also travels in a straight line. The acceleration a m s^-2 of Q is given by a = k - 12t, where k is a constant. The displacement of Q from O is 400 m when t = 10.

2. Find the value of k.

Alan starts walking from a point O, at a constant speed of 4 m s^-1, along a horizontal path. Ben walks along the same path, also starting from O. Ben starts from rest 5 s after Alan and accelerates at 1.2 m s^-2 for 5 s. Ben then continues to walk at a constant speed until he is at the same point, P, as Alan.

(i) Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.

(ii) Find the distance OP.

A cyclist starts from rest at point A and moves in a straight line with acceleration 0.5 m s^-2 for a distance of 36 m. The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point B. The distance AB is 210 m.

(i) Find the total time that the cyclist takes to travel from A to B.

24 s after the cyclist leaves point A, a car starts from rest from point A, with constant acceleration 4 m s^-2, towards B. It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.

(ii) Find the time that it takes from when the cyclist starts until the car overtakes her.

A particle P starts from rest at a point O on a horizontal straight line. P moves along the line with constant acceleration and reaches a point A on the line with a speed of 30 m s^-1. At the instant that P leaves O, a particle Q is projected vertically upwards from the point A with a speed of 20 m s^-1. Subsequently P and Q collide at A. Find

1. the acceleration of P,
2. the distance OA.

A particle X travels in a straight line. The velocity of X at time t s after leaving a fixed point O is denoted by v m/s^-1, where

\(v = -0.1t^3 + 1.8t^2 - 6t + 5.6\).

\(The acceleration of X is zero at t = p and t = q, where p < q.\)

1. Find the value of p and the value of q.
2. It is given that the velocity of X is zero at t = 14.
3. Find the velocities of X at t = p and at t = q, and hence sketch the velocity-time graph for the motion of X for 0 ≤ t ≤ 15.
4. Find the total distance travelled by X between t = 0 and t = 15.

A particle P moves in a straight line starting from a point O and comes to rest 35 s later. At time t s after leaving O, the velocity v m s⁻¹ of P is given by

\(v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,\)

\(v = 2t + 10 \quad 5 \leq t \leq 15,\)

\(v = a + bt^2 \quad 15 \leq t \leq 35,\)

where a and b are constants such that a > 0 and b < 0.

1. Show that the values of a and b are 49 and −0.04 respectively.
2. Sketch the velocity-time graph.
3. Find the total distance travelled by P during the 35 s.

A particle P moves in a straight line. The velocity v m s^-1 at time t s is given by

\(v = 5t(t - 2)\) for \(0 \leq t \leq 4\),

\(v = k\) for \(4 \leq t \leq 14\),

\(v = 68 - 2t\) for \(14 \leq t \leq 20\),

where \(k\) is a constant.

1. Find \(k\).
2. Sketch the velocity-time graph for \(0 \leq t \leq 20\).
3. Find the set of values of \(t\) for which the acceleration of P is positive.
4. Find the total distance travelled by P in the interval \(0 \leq t \leq 20\).

A particle P moves on a straight line. It starts at a point O on the line and returns to O 100 s later. The velocity of P is v m s^-1 at time t s after leaving O, where

\(v = 0.0001t^3 - 0.015t^2 + 0.5t\).

1. Show that P is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\).
2. Find the values of \(v\) at the times for which the acceleration of P is zero, and sketch the velocity-time graph for P's motion for \(0 \leq t \leq 100\).
3. Find the greatest distance of P from O for \(0 \leq t \leq 100\).

A particle starts from rest at a point O and moves in a horizontal straight line. The velocity of the particle is v ms^-1 at time t s after leaving O. For 0 ≤ t < 60, the velocity is given by

\(v = 0.05t - 0.0005t^2\).

The particle hits a wall at the instant when t = 60, and reverses the direction of its motion. The particle subsequently comes to rest at the point A when t = 100, and for 60 < t ≤ 100 the velocity is given by

\(v = 0.025t - 2.5\).

1. Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after it hits the wall.
2. Find the total distance travelled by the particle.
3. Find the maximum speed of the particle and sketch the particle’s velocity-time graph for 0 ≤ t ≤ 100, showing the value of t for which the speed is greatest.

A car driver makes a journey in a straight line from A to B, starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at B. The distance travelled by the car t seconds after leaving A is 0.0000117(400t³ - 3t⁴) metres.

1. Find the distance AB.
2. Find the maximum speed of the car.
3. Find the acceleration of the car
   1. as it starts from A,
   2. as it arrives at B.
4. Sketch the velocity-time graph for the journey.

A particle P moves on the x-axis from the origin O with an initial velocity of \(-20 \text{ ms}^{-1}\). The acceleration \(a \text{ ms}^{-2}\) at time \(t\) s after leaving O is given by \(a = 12 - 2t\).

(a) Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when P is at rest.

(b) Find the total distance travelled by P in the interval \(0 \leq t \leq 12\).

A particle P moves in a straight line starting from a point O and comes to rest 14 s later. At time t s after leaving O, the velocity v m s^-1 of P is given by

\(v = pt^2 - qt \quad 0 \leq t \leq 6,\)

\(v = 63 - 4.5t \quad 6 \leq t \leq 14,\)

where p and q are positive constants.

\(The acceleration of P is zero when t = 2.\)

(a) Given that there are no instantaneous changes in velocity, find p and q.

(b) Sketch the velocity-time graph.

(c) Find the total distance travelled by P during the 14 s.

A particle moves in a straight line and passes through the point A at time \(t = 0\). The velocity of the particle at time \(t\) s after leaving A is \(v\) m s\(^{-1}\), where

\(v = 2t^2 - 5t + 3\).

1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
2. Sketch the velocity-time graph for the first 3 seconds of motion.
3. Find the distance travelled between the two times when the particle is instantaneously at rest.

A particle P moving in a straight line starts from rest at a point O and comes to rest 16 s later. At time t s after leaving O, the acceleration a m s^-2 of P is given by

\(a = 6 + 4t \quad 0 \leq t < 2,\) \(a = 14 \quad 2 \leq t < 4,\) \(a = 16 - 2t \quad 4 \leq t \leq 16.\)

There is no sudden change in velocity at any instant.

1. Find the values of t when the velocity of P is 55 m s^-1.
2. Complete the sketch of the velocity-time diagram.
3. Find the distance travelled by P when it is decelerating.

A particle P moves in a straight line. The velocity v m s^-1 at time t s is given by

\(v = 2t + 1\) for \(0 \leq t \leq 5\),

\(v = 36 - t^2\) for \(5 \leq t \leq 7\),

\(v = 2t - 27\) for \(7 \leq t \leq 13.5\).

(a) Sketch the velocity-time graph for \(0 \leq t \leq 13.5\).

(b) Find the acceleration at the instant when \(t = 6\).

(c) Find the total distance travelled by P in the interval \(0 \leq t \leq 13.5\).

A particle moves in a straight line starting from rest from a point O. The acceleration of the particle at time t s after leaving O is a m/s², where

\(a = 5.4 - 1.62t\).

1. Find the positive value of t at which the velocity of the particle is zero, giving your answer as an exact fraction.
2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
3. Find the total distance travelled during the first ten seconds of the motion.

A particle P moves in a straight line starting from a point O. The velocity v m s^-1 of P at time t s is given by

\(v = 12t - 4t^2\) for \(0 \leq t \leq 2\),

\(v = 16 - 4t\) for \(2 \leq t \leq 4\).

1. Find the maximum velocity of P during the first 2 s.
2. Determine, with justification, whether there is any instantaneous change in the acceleration of P when \(t = 2\).
3. Sketch the velocity-time graph for \(0 \leq t \leq 4\).
4. Find the distance travelled by P in the interval \(0 \leq t \leq 4\).

A particle P moves in a straight line. The velocity v m s^-1 at time t s is given by

\(v = 4 + 0.2t\) for \(0 \leq t \leq 10\),

\(v = -2 + \frac{800}{t^2}\) for \(10 \leq t \leq 20\).

1. Find the acceleration of P during the first 10 s.
2. Find the acceleration of P when \(t = 20\).
3. Sketch the velocity-time graph for \(0 \leq t \leq 20\).
4. Find the total distance travelled by P in the interval \(0 \leq t \leq 20\).

A particle moves in a straight line. At time \(t\) s, the acceleration, \(a \text{ ms}^{-2}\), of the particle is given by \(a = 36 - 6t\). The velocity of the particle is \(27 \text{ ms}^{-1}\) when \(t = 2\).

(a) Find the values of \(t\) when the particle is at instantaneous rest.

(b) Find the total distance the particle travels during the first 12 seconds.