A particle \(A\) has constant velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and a particle \(B\) has constant velocity \(( \mathbf { i } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 6 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\) and \(( - 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) respectively.
Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4 \sqrt { } 2 \mathrm {~m}\).
Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum.
A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW . When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion has magnitude 10 v newtons.
Show that, at the instant when \(v = 20\), the acceleration of the car is \(1.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Find the distance travelled by the car as it accelerates from a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision.
Find the angle through which the path of the ball is deflected by the collision.