Two smooth spheres \(A\) and \(B\) have equal radii and masses 0.4 kg and 0.8 kg respectively. They are moving in opposite directions along the same straight line, with speeds \(3 \mathrm {~ms} ^ { - 1 }\) and 2 \(\mathrm { ms } ^ { - 1 }\) respectively, and collide directly. The coefficient of restitution between \(A\) and \(B\) is 0.8 .
Calculate the speeds of \(A\) and \(B\) after the impact, stating in each case whether the direction of motion has been reversed.
Find the kinetic energy, in J, lost in the impact.
A point of light, \(P\), is moving along a straight line in such a way that, \(t\) seconds after passing through a fixed point \(O\) on the line, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = \frac { 1 } { 2 } t ^ { 2 } - 4 t + 10\). Calculate
the velocity of \(P 6\) seconds after it passes \(O\),
the magnitude of the acceleration of \(P\) when \(t = 1\),
the minimum speed of \(P\),
the distance travelled by \(P\) in the fourth second after it passes \(O\).
A bullet is fired out of a window at a height of 5.2 m above horizontal ground. The initial velocity of the bullet is \(392 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the vertical, where \(\sin \alpha = \frac { 1 } { 20 }\), as shown.
Find
\includegraphics[max width=\textwidth, alt={}, center]{63133ab4-9381-4777-a575-1207219948b7-2_335_490_1343_1419}
the range of times after firing during which the bullet is 15 m or more above ground level,
the greatest height above the ground reached by the bullet,
the horizontal distance travelled by the bullet before it reaches its highest point.
Certain modelling assumptions have been made about the bullet.
State these assumptions and suggest a way in which the model could be refined.
State, with a reason, whether you think this refinement would make a significant difference to the answers.
(2 marks)