Define coefficient of restitution and obtain its value for an elastic collision and a perfectly inelastic collision.

Discuss the following a special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities.

(i) Colliding bodies are identical. Initially at rest,

$$\begin{gathered} \overrightarrow{v_1}=\overrightarrow{u_2} \\ \overrightarrow{v_2}=\overrightarrow{u_k} \end{gathered}$$

Discuss the following special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities.

(i) A very heavy object collides on a lighter object, initially at rest.

$$\begin{gathered} \overrightarrow{v_1}=\overrightarrow{u_1} \\ \overrightarrow{v_2}=k\overrightarrow{u_1} \end{gathered}$$

Discuss the following a special case of elastic collision: A very light object collides on a comparatively much massive object initially at rest. Calculate the ratio of final speed of lighter object to the initial speed of the lighter object.

 A bullet of mass ${\mathrm{m}}_1$ travelling with a velocity $v$ strikes a stationary wooden block of mass  ${\mathrm{m}}_2$ and gets embedded into it. Determine the expression for loss in the kinetic energy of the system. Is this violating the principle of conservation of energy? If not, how can you account for this loss? 

While rotating an object or while opening a door or a water tap we apply a force or forces. Under which conditions is this process easy for us? Why? Define the vector quantity concerned. How does it differ for a single force and for two opposite forces with different lines of action?