Given that the coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\), state the values of \(u\) and \(v\).
Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic.
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Find, for each sphere, its speed and its direction of motion immediately after the contact.
\(4 ~ O\) is a fixed point on a horizontal plane. A particle \(P\) of mass 0.25 kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t \mathrm {~s}\) after release the only horizontal force acting on \(P\) has magnitude
$$\frac { 1 } { 2400 } \left( 144 - t ^ { 2 } \right) \mathrm { N } \quad \text { for } 0 \leqslant t \leqslant 12$$
and
$$\frac { 1 } { 2400 } \left( t ^ { 2 } - 144 \right) \mathrm { N } \text { for } t \geqslant 12 .$$
The force acts in the direction of \(P\) 's motion. \(P\) 's velocity at time \(t \mathrm {~s}\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\).
Sketch the \(( t , v )\) graph for \(0 \leqslant t \leqslant 24\).