Given impulse, find velocity or mass

2. Two particles \(A\) and \(B\) have mass 0.12 kg and 0.08 kg respectively. They are initially at rest on a smooth horizontal table. Particle \(A\) is then given an impulse in the direction \(A B\) so that it moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards \(B\).

1. Find the magnitude of this impulse, stating clearly the units in which your answer is given.
   (2) Immediately after the particles collide, the speed of \(A\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its direction of motion being unchanged.
2. Find the speed of \(B\) immediately after the collision.
3. Find the magnitude of the impulse exerted on \(A\) in the collision.

2. A particle \(P\) is moving with constant acceleration along a straight horizontal line \(A B C\), where \(A C = 24 \mathrm {~m}\). Initially \(P\) is at \(A\) and is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\). After 1.5 s , the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the speed of \(P\) at \(C\) is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg . When \(P\) reaches \(C\), an impulse of magnitude 30 Ns is applied to \(P\) in the direction \(C B\).
2. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant.
   (3)

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.