Past Exam Questions

A particle P moves in a straight line passing through a point O. At time t s, the acceleration, a m s^-2, of P is given by a = 6 - 0.24t. The particle comes to instantaneous rest at time t = 20.

1. Find the value of t at which the particle is again at instantaneous rest.
2. Find the distance the particle travels between the times of instantaneous rest.

A particle starts from rest from a point O and moves in a straight line. The acceleration of the particle at time t after leaving O is a m s^-2, where a = kt^{1/2} for 0 \leq t \leq 9 and where k is a constant. The velocity of the particle at t = 9 is 1.8 m s^-1.

1. Show that k = 0.1.
2. For t > 9, the velocity v m s^-1 of the particle is given by v = 0.2(t - 9)^2 + 1.8.
3. Show that the distance travelled in the first 9 seconds is one tenth of the distance travelled between t = 9 and t = 18.
4. Find the greatest acceleration of the particle during the first 10 seconds of its motion.

A particle P moves in a straight line starting from a point O. At time t s after leaving O, the displacement s m from O is given by \(s = t^3 - 4t^2 + 4t\) and the velocity is \(v\) m s^-1.

1. Find an expression for \(v\) in terms of \(t\).
2. Find the two values of \(t\) for which P is at instantaneous rest.
3. Find the minimum velocity of P.

A particle starts from a fixed origin with velocity 0.4 m s^-1 and moves in a straight line. The acceleration a m s^-2 of the particle t s after it leaves the origin is given by a = k(3t² - 12t + 2), where k is a constant. When t = 1, the velocity of P is 0.1 m s^-1.

1. Show that the value of k is 0.1.
2. Find an expression for the displacement of the particle from the origin in terms of t.
3. Hence verify that the particle is again at the origin at t = 2.

A particle starts from rest and moves in a straight line. The velocity of the particle at time t s after the start is v m s^-1, where

\(v = -0.01t^3 + 0.22t^2 - 0.4t\).

1. Find the two positive values of t for which the particle is instantaneously at rest.
2. Find the time at which the acceleration of the particle is greatest.
3. Find the distance travelled by the particle while its velocity is positive.

A particle starts from a point O and moves in a straight line. The velocity of the particle at time t s after leaving O is v m s^-1, where

\(v = 1.5 + 0.4t \quad \text{for} \quad 0 \leq t \leq 5,\)

\(v = \frac{100}{t^2} - 0.1t \quad \text{for} \quad t \geq 5.\)

1. Find the acceleration of the particle during the first 5 seconds of motion.
2. Find the value of t when the particle is instantaneously at rest.
3. Find the total distance travelled by the particle in the first 10 seconds of motion.

A particle P moves in a straight line starting from a point O. At time t s after leaving O, the velocity, v m s^-1, of P is given by v = (2t - 5)^3.

1. Find the values of t when the acceleration of P is 54 m s^-2.
2. Find an expression for the displacement of P from O at time t s.

A particle P moves in a straight line passing through a point O. At time t s, the velocity of P, v m s^-1, is given by v = qt + rt², where q and r are constants. The particle has velocity 4 m s^-1 when t = 1 and when t = 2.

1. Show that, when t = 0.5, the acceleration of P is 4 m s^-2.
2. Find the values of t when P is at instantaneous rest.
3. The particle is at O when t = 3. Find the distance of P from O when t = 0.

A particle moves in a straight line. Its displacement t s after leaving a fixed point O on the line is s m, where \(s = 2t^2 - \frac{80}{3}t^{3/2}\).

1. Find the time at which the acceleration of the particle is zero.
2. Find the displacement and velocity of the particle at this instant.

A racing car is moving in a straight line. The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after the car starts from rest is given by

\(a = 15t - 3t^2 \quad \text{for} \; 0 \leq t \leq 5,\)

\(a = -\frac{625}{t^2} \quad \text{for} \; 5 < t \leq k,\)

where \(k\) is a constant.

1. Find the maximum acceleration of the car in the first five seconds of its motion. [3]
2. Find the distance of the car from its starting point when \(t = 5\). [3]
3. The car comes to rest when \(t = k\). Find the value of \(k\). [5]

A particle P moves in a straight line. At time t s, the displacement of P from O is s m and the acceleration of P is a m s^-2, where a = 6t - 2. When t = 1, s = 7 and when t = 3, s = 29.

1. Find the set of values of t for which the particle is decelerating.
2. Find s in terms of t.
3. Find the time when the velocity of the particle is 10 m s^-1.

A particle P moves in a straight line, starting from a point O. At time t s after leaving O, the velocity of P, v m s^-1, is given by v = 4t² - 8t + 3.

1. Find the two values of t at which P is at instantaneous rest.
2. Find the distance travelled by P between these two times.

A particle P starts at rest and moves in a straight line from a point O. At time t s after leaving O, the velocity of P, v m/s, is given by \(v = bt + ct^{\frac{3}{2}}\), where b and c are constants. P has velocity 8 m/s when \(t = 4\) and has velocity 13.5 m/s when \(t = 9\).

1. Show that \(b = 3\) and \(c = -0.5\).
2. Find the acceleration of P when \(t = 1\).
3. Find the positive value of t when P is at instantaneous rest and find the distance of P from O at this instant.
4. Find the speed of P at the instant it returns to O.

A particle P moves in a straight line. It starts at a point O on the line and at time t s after leaving O it has a velocity v m s^-1, where v = 6t^2 - 30t + 24.

1. Find the set of values of t for which the acceleration of the particle is negative.
2. Find the distance between the two positions at which P is at instantaneous rest.
3. Find the two positive values of t at which P passes through O.

A particle P starts from rest at a point O of a straight line and moves along the line. The displacement of the particle at time t s after leaving O is x m, where

\(x = 0.08t^2 - 0.0002t^3\).

1. Find the value of t when P returns to O and find the speed of P as it passes through O on its return.
2. For the motion of P until the instant it returns to O, find
   1. the total distance travelled,
   2. the average speed.

A particle P moves along a straight line for 100 s. It starts at a point O and at time t seconds after leaving O the velocity of P is v m/s, where

\(v = 0.00004t^3 - 0.006t^2 + 0.288t\).

1. Find the values of t at which the acceleration of P is zero.
2. Find the displacement of P from O when t = 100.

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

\(v = 0.6t - 0.03t^2\).

1. Verify that, when \(t = 5\), the particle is 6.25 m from O. Find the acceleration of the particle at this time.
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.

A particle P moves in a straight line. At time t seconds after starting from rest at the point O on the line, the acceleration of P is a m/s², where a = 0.075t² - 1.5t + 5.

1. Find an expression for the displacement of P from O in terms of t.
2. Hence find the time taken for P to return to the point O.

A particle P starts from rest and moves in a straight line for 18 seconds. For the first 8 seconds of the motion P has constant acceleration 0.25 m/s². Subsequently P's velocity, v m/s^-1 at time t seconds after the motion started, is given by

\(v = -0.1t^2 + 2.4t - k\),

where \(8 \leq t \leq 18\) and \(k\) is a constant.

1. Find the value of \(v\) when \(t = 8\) and hence find the value of \(k\).
2. Find the maximum velocity of P.
3. Find the displacement of P from its initial position when \(t = 18\).

The diagram shows the velocity-time graph for the motion of a particle P which moves on a straight line BAC. It starts at A and travels to B taking 5 s. It then reverses direction and travels from B to C taking 10 s. For the first 3 s of P's motion its acceleration is constant. For the remaining 12 s the velocity of P is v m s^-1 at time t s after leaving A, where

\(v = -0.2t^2 + 4t - 15\) for \(3 \leq t \leq 15\).

1. Find the value of v when t = 3 and the magnitude of the acceleration of P for the first 3 s of its motion.
2. Find the maximum velocity of P while it is moving from B to C.
3. Find the average speed of P,
   1. while moving from A to B,
   2. for the whole journey.