Past Exam Questions

A particle P travels in a straight line. It passes through the point O of the line with velocity 5 m s^-1 at time t = 0, where t is in seconds. P's velocity after leaving O is given by

(0.002t³ - 0.12t² + 1.8t + 5) m s^-1.

The velocity of P is increasing when 0 < t < T₁ and when t > T₂, and the velocity of P is decreasing when T₁ < t < T₂.

1. Find the values of T₁ and T₂ and the distance OP when t = T₂.
2. Find the velocity of P when t = T₂ and sketch the velocity-time graph for the motion of P.

A particle starts at a point O and moves along a straight line. Its velocity t s after leaving O is \((1.2t - 0.12t^2)\) m s^-1. Find the displacement of the particle from O when its acceleration is 0.6 m s^-2.

A vehicle is moving in a straight line. The velocity \(v\) m s^-1 at time \(t\) s after the vehicle starts is given by

\(v = A(t - 0.05t^2) \quad \text{for} \; 0 \leq t \leq 15,\)

\(v = \frac{B}{t^2} \quad \text{for} \; t \geq 15,\)

where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m.

1. Find the value of \(A\) and show that \(B = 3375\).
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\).
3. Find the speed of the vehicle when it has travelled a total distance of 315 m.

A motorcyclist starts from rest at A and travels in a straight line. For the first part of the motion, the motorcyclist’s displacement x metres from A after t seconds is given by x = 0.6t² - 0.004t³.

1. Show that the motorcyclist’s acceleration is zero when t = 50 and find the speed V m s^-1 at this time.
2. For t ≥ 50, the motorcyclist travels at constant speed V m s^-1. Find the value of t for which the motorcyclist’s average speed is 27.5 m s^-1.

A particle P starts from rest at the point A at time t = 0, where t is in seconds, and moves in a straight line with constant acceleration a m s^-2 for 10 s. For 10 ≤ t ≤ 20, P continues to move along the line with velocity v m s^-1, where v = \(\frac{800}{t^2} - 2\). Find

1. the speed of P when t = 10, and the value of a,
2. the value of t for which the acceleration of P is -a m s^-2,
3. the displacement of P from A when t = 20.

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

\(a = 0.3t^{\frac{1}{2}}\) for \(0 \leq t \leq 4\),

\(a = -kt^{-\frac{3}{2}}\) for \(4 < t \leq T\),

where k and T are constants.

1. Find the velocity of P at \(t = 4\).
2. It is given that there is no change in the velocity of P at \(t = 4\) and that the velocity of P at \(t = 16\) is \(0.3 \text{ m/s}\). Show that \(k = 2.6\) and find an expression, in terms of t, for the velocity of P for \(4 \leq t \leq T\).
3. Given that P comes to instantaneous rest at \(t = T\), find the exact value of T.
4. Find the total distance travelled between \(t = 0\) and \(t = T\).

A particle P travels in a straight line from A to D, passing through the points B and C. For the section AB the velocity of the particle is \((0.5t - 0.01t^2)\) m s\(^{-1}\), where \(t\) is the time after leaving A.

1. Given that the acceleration of P at B is 0.1 m s\(^{-2}\), find the time taken for P to travel from A to B.
2. The acceleration of P from B to C is constant and equal to 0.1 m s\(^{-2}\). Given that P reaches C with speed 14 m s\(^{-1}\), find the time taken for P to travel from B to C.
3. P travels with constant deceleration 0.3 m s\(^{-2}\) from C to D. Given that the distance CD is 300 m, find
   1. the speed with which P reaches D,
   2. the distance AD.

An object P travels from A to B in a time of 80 s. The diagram shows the graph of v against t, where v m s^-1 is the velocity of P at time t s after leaving A. The graph consists of straight line segments for the intervals 0 ≤ t ≤ 10 and 30 ≤ t ≤ 80, and a curved section whose equation is v = -0.01t² + 0.5t - 1 for 10 ≤ t ≤ 30. Find

1. the maximum velocity of P,
2. the distance AB.

The velocity of a particle at time t seconds after it starts from rest is v m/s, where \(v = 1.25t - 0.05t^2\). Find

1. the initial acceleration of the particle,
2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \text{ m/s}^2\).

A motorcyclist starts from rest at A and travels in a straight line until he comes to rest again at B. The velocity of the motorcyclist t seconds after leaving A is v m s^-1, where v = t - 0.01t^2. Find

1. the time taken for the motorcyclist to travel from A to B,
2. the distance AB.

A particle P starts from rest at O and travels in a straight line. Its velocity v m s^-1 at time t s is given by v = 8t - 2t^2 for 0 ≤ t ≤ 3, and v = \frac{54}{t^2} for t > 3. Find

1. the distance travelled by P in the first 3 seconds,
2. an expression in terms of t for the displacement of P from O, valid for t > 3,
3. the value of v when the displacement of P from O is 27 m.

A particle P moves along the x-axis in the positive direction. The velocity of P at time t s is 0.03t² m s⁻¹. When t = 5 the displacement of P from the origin O is 2.5 m.

1. Find an expression, in terms of t, for the displacement of P from O.
2. Find the velocity of P when its displacement from O is 11.25 m.

A particle starts from rest at the point A and travels in a straight line until it reaches the point B. The velocity of the particle t seconds after leaving A is v m s^-1, where v = 0.009t^2 - 0.0001t^3. Given that the velocity of the particle when it reaches B is zero, find

1. the time taken for the particle to travel from A to B,
2. the distance AB,
3. the maximum velocity of the particle.

A particle P moves in a straight line that passes through the origin O. The velocity of P at time t seconds is v m s^-1, where v = 20t - t^3. At time t = 0 the particle is at rest at a point whose displacement from O is -36 m.

1. Find an expression for the displacement of P from O in terms of t.
2. Find the displacement of P from O when t = 4.
3. Find the values of t for which the particle is at O.

A particle moves in a straight line. Its displacement t seconds after leaving the fixed point O is x metres, where \(x = \frac{1}{2}t^2 + \frac{1}{30}t^3\). Find

1. the speed of the particle when \(t = 10\),
2. the value of \(t\) for which the acceleration of the particle is twice its initial acceleration.

A particle P starts to move from a point O and travels in a straight line. At time t s after P starts to move its velocity is v m s^-1, where v = 0.12t - 0.0006t².

1. Verify that P comes to instantaneous rest when t = 200, and find the acceleration with which it starts to return towards O.
2. Find the maximum speed of P for 0 ≤ t ≤ 200.
3. Find the displacement of P from O when t = 200.
4. Find the value of t when P reaches O again.

A particle P moves in a straight line through a point O. The velocity v ms^-1 of P, at time t s after passing O, is given by

\(v = \frac{9}{4} + \frac{b}{(t+1)^2} - ct^2,\)

where b and c are positive constants. At t = 5, the velocity of P is zero and its acceleration is \(-\frac{13}{12}\) ms^-2.

\((a) Show that b = 9 and find the value of c.\)

\((b) Given that the velocity of P is zero only at t = 5, find the distance travelled in the first 10 seconds of motion.\)

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

\(v = 0.5t\) for \(0 \leq t \leq 10\),

\(v = 0.25t^2 - 8t + 60\) for \(10 \leq t \leq 20\).

(a) Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\).

(b) Find the total distance covered by P in the interval \(0 \leq t \leq 20\).