3 A particle of mass 4 kg moves along the \(x\)-axis. At time \(t\) seconds the particle is \(x \mathrm {~m}\) from the origin O and has velocity \(v \mathrm {~ms} ^ { - 1 }\). A driving force of magnitude \(20 t \mathrm { t } ^ { - t } \mathrm {~N}\) is applied in the positive \(x\) direction. Initially \(v = 2\) and the particle is at O .

1. Find, in either order, the impulse of the force over the first 3 seconds and the velocity of the particle after 3 seconds. From time \(t = 3\) a resistive force of magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\) and the driving force are applied until the particle comes to rest.
2. Show that, after the resistive force is applied, the only time at which the resultant force on the particle is zero is when \(t = \ln 40\). Hence find the maximum velocity of the particle during the motion.
3. Given that the time \(T\) seconds at which the particle comes to rest is given by the equation \(T = \sqrt { 121 - 80 \mathrm { e } ^ { - T } ( 1 + T ) }\), without solving the equation deduce that \(T \approx 11\).
4. Use a numerical method to find \(T\) correct to 4 decimal places.