In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
the acceleration of the aircraft,
the distance \(B C\).
Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
the speed of \(A\) immediately after the collision,
the magnitude of the impulse exerted on \(B\) in the collision.
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A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(A C\). The bead \(B\) is vertically below \(C\) and \(\angle B A C = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac { 3 } { 4 }\), find