OCR Further Mechanics (Further Mechanics) 2017 Specimen

1 A body, \(P\), of mass 2 kg moves under the action of a single force \(\mathbf { F } \mathrm { N }\). At time \(t \mathrm {~s}\), the velocity of the body is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( t ^ { 2 } - 3 \right) \mathbf { i } + \frac { 5 } { 2 t + 1 } \mathbf { j } \text { for } t \geq 2 .$$

1. Obtain \(\mathbf { F }\) in terms of \(t\).
2. Calculate the rate at which the force \(\mathbf { F }\) is working at \(t = 4\).
3. By considering the change in kinetic energy of \(P\), calculate the work done by the force \(\mathbf { F }\) during the time interval \(2 \leq t \leq 4\).

3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F N\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4 .$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).