- A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction of \(x\) increasing.
When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \frac { 5 } { 2 } ( 5 - v )\)
- show that \(v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)\)
- state the limiting value of \(v\) as \(t\) increases.
At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
- Show that \(d = 2 \ln 2 - 1\)