16 A particle, of mass 400 grams, is initially at rest at the point \(O\).
The particle starts to move in a straight line so that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by
$$v = 6 t ^ { 2 } - 12 t ^ { 3 } \text { for } t > 0$$
16
- Find an expression, in terms of \(t\), for the force acting on the particle.
[0pt]
[3 marks]
16 - Find the time when the particle next passes through \(O\).
[0pt]
[5 marks]
In this question use \(\boldsymbol { g } = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope.
The rope passes over a light smooth pulley, as shown in the diagram.
The rope between the pulley and the van is horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-20_515_766_685_607}
Initially, the van is at rest and the crate rests on the lower level. The rope is taut.
The van moves away from the pulley to lift the crate from the lower level.
The van's engine produces a constant driving force of 5000 N .
A constant resistance force of magnitude 780 N acts on the van.
Assume there is no resistance force acting on the crate.