Mechanics — Variant 2

Small smooth spheres $A$ and $B$, of equal radii and of masses $5$ kg and $3$ kg respectively, lie on a smooth horizontal plane. Initially $B$ is at rest and $A$ is moving towards $B$ with speed $8.5$ $\text{m s}^{-1}$. The spheres collide and after the collision $A$ continues to move in the same direction but with a quarter of the speed of $B$.

Find the speed of $B$ after the collision.

Find the loss of kinetic energy of the system due to the collision.

Two particles $A$ and $B$, of masses $2.4$ kg and $1.2$ kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. $A$ is held at a distance of $2.1$ m above a horizontal plane and $B$ is $1.5$ m above the plane. The particles hang vertically and are released from rest. In the subsequent motion $A$ reaches the plane and does not rebound and $B$ does not reach the pulley.

Show that the tension in the string before $A$ reaches the plane is $16$ N and find the magnitude of the acceleration of the particles before $A$ reaches the plane.

Find the greatest height of $B$ above the plane.

A particle $A$, moving along a straight horizontal track with constant speed $8$ $\text{m s}^{-1}$, passes a fixed point $O$. Four seconds later, another particle $B$ passes $O$, moving along a parallel track in the same direction as $A$. Particle $B$ has speed $20$ $\text{m s}^{-1}$ when it passes $O$ and has a constant deceleration of $2$ $\text{m s}^{-2}$. $B$ comes to rest when it returns to $O$.

Find expressions, in terms of $t$, for the displacement from $O$ of each particle $t$ seconds after $B$ passes $O$.

Find the values of $t$ when the particles are the same distance from $O$.

On the given axes, sketch the displacement-time graphs for both particles, for values of $t$ from $0$ to $20$.

A car of mass $900$ kg is moving up a hill inclined at $\sin^{-1} 0.12$ to the horizontal. The initial speed of the car is $11$ $\text{m s}^{-1}$. After $12$ s, the car has travelled $150$ m up the hill and has speed $16$ $\text{m s}^{-1}$. The engine of the car is working at a constant rate of $24$ kW.

Find the work done against the resistive forces during the $12$ s. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of $(1520 + 4v)$ N when the speed of the car is $v$ $\text{m s}^{-1}$. The car travels at a constant speed with the engine working at a constant rate of $32$ kW.

Find this speed.

A particle $P$ moves in a straight line. The velocity $v$ $\text{m s}^{-1}$ at time $t$ seconds is given by $v = 0.5t$ for $0 \leqslant t \leqslant 10$, $v = 0.25t^{2} - 8t + 60$ for $10 \leqslant t \leqslant 20$.

Show that there is an instantaneous change in the acceleration of the particle at $t = 10$.

Find the total distance covered by $P$ in the interval $0 \leqslant t \leqslant 20$.