A bullet of mass 0.05 kg is fired with speed \(u \mathrm {~ms} ^ { - 1 }\) from a gun, which recoils at a speed of \(0.008 u \mathrm {~ms} ^ { - 1 }\) in the opposite direction to that in which the bullet is fired.
- Find the mass of the gun.
- Find, in terms of \(u\), the kinetic energy given to the bullet and to the gun at the instant of firing.
- If the total kinetic energy created in firing the gun is 5100 J , find the value of \(u\).
- The acceleration of a particle \(P\) at time \(t \mathrm {~s}\) is \(\mathbf { a } \mathrm { ms } ^ { - 2 }\), where \(\mathbf { a } = 4 \mathrm { e } ^ { t } \mathbf { i } - \mathrm { e } ^ { t } \mathbf { j }\). When \(t = 0 , P\) has velocity \(4 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
- Find the speed of \(P\) when \(t = 2\).
- Find the time at which the direction of motion of \(P\) is parallel to the vector \(5 \mathbf { i } - \mathbf { j }\).
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A uniform plank \(A B\), of mass 3 kg and length 2 m , rests in equilibrium with the point \(P\) in contact with a smooth cylinder. The end \(B\) rests on a rough horizontal surface and the coefficient of friction between the plank and the surface is \(\frac { 1 } { 3 } . A B\) makes an angle of \(60 ^ { \circ }\) with the horizontal.
If the plank is in limiting equilibrium in this position, find