5. Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m \mathrm {~kg}\) and km kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude 7 km Ns.
Q. 5 continued on next page ... \section*{MECHANICS 1 (A) TEST PAPER 7 Page 2}

1. continued ...
   1. Find the speed of \(B\) after the impact.
   2. Show that the speed of \(A\) immediately after the collision is \(( 7 k - 5 ) \mathrm { ms } ^ { - 1 }\) and deduce that the direction of \(A\) 's motion is reversed.
      \(B\) is now given a further impulse of magnitude \(m u \mathrm { Ns }\), as a result of which a second collision between it and \(A\) occurs.
   3. Show that \(u > k ( 7 k - 1 )\).
   \includegraphics[max width=\textwidth, alt={}]{6fbe12d6-9a46-4602-a7b9-63d50b02ff28-2_422_787_815_340}
   The velocity-time graph illustrates the motion of a particle which accelerates from rest to \(8 \mathrm {~ms} ^ { - 1 }\) in \(x\) seconds and then to \(24 \mathrm {~ms} ^ { - 1 }\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to \(12 \mathrm {~ms} ^ { - 1 }\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. Given that the total distance travelled by the particle is 496 m ,
2. show that \(2 x + 21 y = 195\). Given also that the average speed of the particle during its motion is \(15 \cdot 5 \mathrm {~ms} ^ { - 1 }\),
3. show that \(x + 2 y = 21\).
4. Hence find the values of \(x\) and \(y\),
5. Write down the acceleration for each section of the motion.