1 Two sledges P and Q, with their loads, have masses of 200 kg and 250 kg respectively and are sliding on a horizontal surface against negligible resistance. There is an inextensible light rope connecting the sledges that is horizontal and parallel to the direction of motion.
Fig. 1 shows the initial situation with both sledges travelling with a velocity of \(5 \mathbf { i m ~ } \mathbf { m } ^ { - 1 }\).
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\caption{Fig. 1}
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A mechanism on Q jerks the rope so that there is an impulse of \(250 \mathbf { i N s }\) on P .
- Show that the new velocity of \(P\) is \(6.25 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\) and find the new velocity of \(Q\).
There is now a direct collision between the sledges and after the impact P has velocity \(4.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).
- Show that the velocity of Q becomes \(5.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Calculate the coefficient of restitution in the collision.
Before the rope becomes taut again, the velocity of P is increased so that it catches up with Q . This is done by throwing part of the load from sledge P in the \(- \mathbf { i }\) direction so that P 's velocity increases to \(5.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). The part of the load thrown out is an object of mass 20 kg .
- Calculate the speed of separation of the object from P .
When the sledges directly collide again they are held together and move as a single object.
- Calculate the common velocity of the pair of sledges, giving your answer correct to 3 significant figures.
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\caption{not to scale he lengths are}
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Fig. 2
Fig. 2 shows a stand on a horizontal floor and horizontal and vertical coordinate axes \(\mathrm { O } x\) and \(\mathrm { O } y\). The stand is modelled as
- a thin uniform rectangular base PQRS, 30 cm by 40 cm with mass 15 kg ; the sides QR and PS are parallel to \(\mathrm { O } x\),
- a thin uniform vertical rod of length 200 cm and mass 3 kg that is fixed to the base at O , the mid-point of PQ and the origin of coordinates,
- a thin uniform top rod AB of length 50 cm and mass \(2 \mathrm {~kg} ; \mathrm { AB }\) is parallel to \(\mathrm { O } x\).
Coordinates are referred to the axes shown in the figure.