A hoop of radius $$r$$ and mass $$m$$ rotating with an angular velocity $${\omega _0}$$ is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop when it cases to slip?

This question has statement $${\rm I}$$ and statement $${\rm I}$$$${\rm I}$$. Of the four choices given after the statements, choose the one that best describes the two statements.

Statement - $${\rm I}$$: A point particle of mass $$m$$ moving with speed $$\upsilon $$ collides with stationary point particle of mass $$M.$$ If the maximum energy loss possible is given as $$f\left( {{1 \over 2}m{v^2}} \right)$$, then $$f = \left( {{m \over {M + m}}} \right).$$

Statement - $${\rm II}$$: Maximum energy loss occurs when the particles get stuck together as a result of the collision.

A projectile is given an initial velocity of $$\left( {\widehat i + 2\widehat j} \right)$$ m/s, where $${\widehat i}$$ is along the ground and $${\widehat j}$$ is along the vertical. If g = 10 m/s², the equation of its trajectory is:

Let [$${\varepsilon _0}$$] denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then: