- A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.
In a model of the motion of \(P\)
$$a = 4 - 3 v$$
When \(t = 0 , v = 0\)
- Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found.
When \(t = 0 , P\) is at the origin \(O\)
- Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.