Time to reach given speed

3 A car of mass 1000 kg is moving on a straight horizontal road. The driving force of the car is \(\frac { 28000 } { v } \mathrm {~N}\) and the resistance to motion is \(4 \nu \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car \(t\) seconds after it passes a fixed point on the road.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 7000 - v ^ { 2 } } { 250 v }\). The car passes points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
2. Find the time taken for the car to travel from \(A\) to \(B\).

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude \(F\) newtons acts on the particle in a constant direction for 0.1 seconds. At time \(t\) seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of \(t\) when the speed of the particle is \(2 \mathrm {~ms} ^ { - 1 }\).
(4 marks)

1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)