Past Exam Questions

A particle P moves on a straight line, starting from rest at a point O of the line. The time after P starts to move is t s, and the particle moves along the line with constant acceleration \(\frac{1}{4} \text{ m s}^{-2}\) until it passes through a point A at time \(t = 8\). After passing through A the velocity of P is \(\frac{1}{2} t^{2/3} \text{ m s}^{-1}\).

1. Find the acceleration of P immediately after it passes through A. Hence show that the acceleration of P decreases by \(\frac{1}{12} \text{ m s}^{-2}\) as it passes through A.
2. Find the distance moved by P from \(t = 0\) to \(t = 27\).

A vehicle starts from rest at a point O and moves in a straight line. Its speed \(v\) m s\(^{-1}\) at time \(t\) seconds after leaving O is defined as follows.

For \(0 \leq t \leq 60\), \(v = k_1 t - 0.005t^2\),

for \(t \geq 60\), \(v = \frac{k_2}{\sqrt{t}}\).

The distance travelled by the vehicle during the first 60 s is 540 m.

1. Find the value of the constant \(k_1\) and show that \(k_2 = 12\sqrt{60}\).
2. Find an expression in terms of \(t\) for the total distance travelled when \(t \geq 60\).
3. Find the speed of the vehicle when it has travelled a total distance of 1260 m.

A particle P moves in a straight line. P starts from rest at O and travels to A where it comes to rest, taking 50 seconds. The speed of P at time t seconds after leaving O is v m/s^-1, where v is defined as follows.

\(For 0 ≤ t ≤ 5, v = t - 0.1t²,\)

for 5 ≤ t ≤ 45, v is constant,

\(for 45 ≤ t ≤ 50, v = 9t - 0.1t² - 200.\)

(i) Find the distance travelled by P in the first 5 seconds.

(ii) Find the total distance from O to A, and deduce the average speed of P for the whole journey from O to A.

A particle moves in a straight line starting from rest. The displacement s m of the particle from a fixed point O on the line at time t s is given by

\(s = t^{\frac{5}{2}} - \frac{15}{4} t^{\frac{3}{2}} + 6\).

Find the value of s when the particle is again at rest.

A particle P starts from rest at a point O and moves in a straight line. P has acceleration 0.6t m s⁻² at time t seconds after leaving O, until t = 10.

1. Find the velocity and displacement from O of P when t = 10.

\(After t = 10, P has acceleration −0.4t m s⁻² until it comes to rest at a point A.\)

2. Find the distance OA.

An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at O and has speed 90 m s^-1 at the instant it takes off. While the aeroplane is on the runway at time t seconds after leaving O, its acceleration is (1.5 + 0.012t) m s^-2. Find

1. the value of t at the instant the aeroplane takes off,
2. the distance travelled by the aeroplane on the runway.

A particle moves in a straight line. Its velocity t seconds after leaving a fixed point O on the line is v m s^-1, where v = 0.2t + 0.006t². For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that t = 25,
2. find the displacement of the particle from O.

A particle P starts to move from a point O and travels in a straight line. The velocity of P is \(k(60t^2 - t^3)\) m s^-1 at time t s after leaving O, where k is a constant. The maximum velocity of P is 6.4 m s^-1.

1. Show that \(k = 0.0002\).

P comes to instantaneous rest at a point A on the line. Find

2. the distance OA,
3. the magnitude of the acceleration of P at A,
4. the speed of P when it subsequently passes through O.

A car travels along a straight road with constant acceleration \(a \text{ m s}^{-2}\). It passes through points \(A, B\) and \(C\); the time taken from \(A\) to \(B\) and from \(B\) to \(C\) is 5 s in each case. The speed of the car at \(A\) is \(u \text{ m s}^{-1}\) and the distances \(AB\) and \(BC\) are 55 m and 65 m respectively. Find the values of \(a\) and \(u\).

A particle P travels from a point O along a straight line and comes to instantaneous rest at a point A. The velocity of P at time t s after leaving O is v m s^-1, where v = 0.027(10t² - t³). Find

1. the distance OA,
2. the maximum velocity of P while moving from O to A.

A particle P moves in a straight line, starting from the point O with velocity 2 m s^-1. The acceleration of P at time t s after leaving O is 2t^2/3 m s^-2.

1. Show that t^5/3 = 5/6 when the velocity of P is 3 m s^-1.
2. Find the distance of P from O when the velocity of P is 3 m s^-1.

A particle P starts at the point O and travels in a straight line. At time t seconds after leaving O the velocity of P is v m s^-1, where v = 0.75t² - 0.0625t³. Find

1. the positive value of t for which the acceleration is zero,
2. the distance travelled by P before it changes its direction of motion.

A particle P moves in a straight line. It starts from rest at A and comes to rest instantaneously at B. The velocity of P at time t seconds after leaving A is v m/s, where v = 6t^2 - kt^3 and k is a constant.

1. Find an expression for the displacement of P from A in terms of t and k.
2. Find an expression for t in terms of k when P is at B.

Given that the distance AB is 108 m, find

3. the value of k,
4. the maximum value of v when the particle is moving from A towards B.

A tractor travels in a straight line from a point A to a point B. The velocity of the tractor is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving A.

(i) The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for

1. the distance \(AB\),
2. the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).

(ii) The actual velocity of the tractor is given by \(v = 0.04t - 0.00005t^2\) for \(0 \leq t \leq 800\).

1. Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i).

For the interval \(0 \leq t \leq 400\), the approximate velocity of the tractor in part (i) is denoted by \(v_1 \text{ m s}^{-1}\).

2. Express \(v_1\) in terms of \(t\) and hence show that \(v_1 - v = 0.00005(t - 200)^2 - 1\).
3. Deduce that \(-1 \leq v_1 - v \leq 1\).

A particle moves in a straight line starting from rest from a point O. The acceleration of the particle at time t seconds after leaving O is a m/s², where a = 4t^{\frac{1}{2}}.

\((a) Find the speed of the particle when t = 9.\)

(b) Find the time after leaving O at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal.

A particle P moves in a straight line. It starts from a point O on the line with velocity 1.8 m s^-1. The acceleration of P at time t s after leaving O is 0.8t^-0.75 m s^-2. Find the displacement of P from O when t = 16.

A particle P starts from a point O and moves along a straight line. P's velocity t s after leaving O is v m s^-1, where

\(v = 0.16t^{\frac{3}{2}} - 0.016t^2\).

P comes to rest instantaneously at the point A.

1. Verify that the value of t when P is at A is 100.
2. Find the maximum speed of P in the interval \(0 < t < 100\).
3. Find the distance OA.
4. Find the value of t when P passes through O on returning from A.

A particle travels in a straight line from A to B in 20 s. Its acceleration t seconds after leaving A is a m s^-2, where a = \frac{3}{160}t^2 - \frac{1}{800}t^3. It is given that the particle comes to rest at B.

1. Show that the initial speed of the particle is zero.
2. Find the maximum speed of the particle.
3. Find the distance AB.

A particle travels in a straight line from a point P to a point Q. Its velocity t seconds after leaving P is v m s^-1, where v = 4t - \frac{1}{16}t^3. The distance PQ is 64 m.

1. Find the time taken for the particle to travel from P to Q.
2. Find the set of values of t for which the acceleration of the particle is positive.

A particle travels along a straight line. It starts from rest at a point A on the line and comes to rest again, 10 seconds later, at another point B on the line. The velocity t seconds after leaving A is

\(0.72t^2 - 0.096t^3\) for \(0 \leq t \leq 5\),

\(2.4t - 0.24t^2\) for \(5 \leq t \leq 10\).

1. Show that there is no instantaneous change in the acceleration of the particle when \(t = 5\).
2. Find the distance \(AB\).