1. A car of mass 500 kg moves along a straight horizontal road.

The engine of the car produces a constant driving force of 1800 N .
The car accelerates from rest from the fixed point \(O\) at time \(t = 0\) and at time \(t\) seconds the car is \(x\) metres from \(O\), moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car has magnitude \(2 v ^ { 2 } \mathrm {~N}\). At time \(T\) seconds, the car is at the point \(A\), moving with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(T = \frac { 25 } { 6 } \ln 2\)
2. Show that the distance from \(O\) to \(A\) is \(125 \ln \frac { 9 } { 8 } \mathrm {~m}\).