4 The diagram shows the path of a particle P of mass 2 kg as it moves from the origin O to C via A and B . The lengths of the sections \(\mathrm { OA } , \mathrm { AB }\) and BC are given in the diagram. The units of the axes are metres.
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P , starting from O , moves along the path indicated in the diagram to C under the action of a constant force of magnitude \(T \mathrm {~N}\) acting in the positive \(x\)-direction. As P moves, it does \(R \mathrm {~J}\) of work for every metre travelled against resistances to motion.
It is given that
- the speed of P at O is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
- the speed of P at A is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
- the speed of P at C is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
You should assume that both \(x\) - and \(y\)-axes lie in a horizontal plane.
- By considering the entire path of P from O to C , show that
$$20 \mathrm {~T} - 30 \mathrm { R } = 108 .$$
- By formulating a second equation, determine the values of \(T\) and \(R\).
- It is now given that the \(x\)-axis is horizontal, and the \(y\)-axis is directed vertically upwards. By considering the kinetic energy of P at B , show that the motion as described above is impossible.