1 A sledge and a child sitting on it have a combined mass of 29.5 kg . The sledge slides on horizontal ice with negligible resistance to its movement.
- While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution in the collision is 0.8 . After the impact the speeds of the sledge and the ball are \(V _ { 1 } \mathrm {~ms} ^ { - 1 }\) and \(V _ { 2 } \mathrm {~ms} ^ { - 1 }\) respectively.
Calculate \(V _ { 1 }\) and \(V _ { 2 }\) and state the direction in which the ball is travelling after the impact. [7]
- While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The snowball sticks to the sledge.
(A) Calculate the velocity with which the combined sledge and snowball start to move.
(B) The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. - In another situation, the sledge is travelling over the ice at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with 10.5 kg of snow on it (giving a total mass of 40 kg ). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the snowball and sledge are separating at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown.
Calculate \(V\).
\begin{figure}[h]
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\caption{Fig. 2}
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Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\), \(\mathrm { BD } , \mathrm { BE } , \mathrm { CE }\) and DE . [The triangles \(\mathrm { ABD } , \mathrm { BDE }\) and BCE are all equilateral.]
The rods \(\mathrm { AB } , \mathrm { BC }\) and DE are horizontal.
The rods are freely pin-jointed to each other at A, B, C, D and E.
The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD . The pin-joint at D rests on this plane.
The following external forces act on the framework: a vertical load of \(L \mathrm {~N}\) at C ; the normal reaction force \(R \mathrm {~N}\) of the plane on the framework at D ; the horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), respectively, acting at A . - Write down equations for the horizontal and vertical equilibrium of the framework.
- By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt { 3 } L\) and \(Y = 0\).
- Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods.
- Show that the internal force in the rod AD is zero.
- Find the forces internal to \(\mathrm { AB } , \mathrm { CE }\) and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.]
- Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust.