An object, moving with a speed of $6.25 \mathrm{~m}/\mathrm{s}$, is decelerated at a rate given by :
$$\frac{\mathrm{dv}}{\mathrm{dt}} = -2.5\sqrt{\mathrm{v}}$$
where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be:
(1) 2 s
(2) 4 s
(3) 8 s
(4) 1 s

A particle is moving with speed $v = b \sqrt { x }$ along positive $x$ - axis. Calculate the speed of the particle at time $t = \tau$ (assume that the particle is at origin at $t = 0$ )
(1) $b ^ { 2 } \tau$
(2) $\frac { b ^ { 2 } \tau } { \sqrt { 2 } }$
(3) $\frac { b ^ { 2 } \tau } { 2 }$
(4) $\frac { b ^ { 2 } \tau } { 4 }$

A ball is thrown upward with an initial velocity $V _ { 0 }$ from the surface of the earth. The motion of the ball is affected by a drag force equal to $\mathrm { m } \gamma v ^ { 2 }$ (where m is mass of the ball, $v$ is its instantaneous velocity and $\gamma$ is a constant). Time taken by the ball to rise to its zenith is:
(1) $\frac { 1 } { \sqrt { \gamma g } } \ln \left( 1 + \sqrt { \frac { \gamma } { g } } \mathrm {~V} _ { 0 } \right)$
(2) $\frac { 1 } { \sqrt { \gamma g } } \tan ^ { - 1 } \left( \sqrt { \frac { \gamma } { g } } \mathrm {~V} _ { 0 } \right)$
(3) $\frac { 1 } { \sqrt { \gamma \mathrm {~g} } } \sin ^ { - 1 } \left( \sqrt { \frac { \gamma } { \mathrm {~g} } } \mathrm {~V} _ { 0 } \right)$
(4) $\frac { 1 } { \sqrt { 2 \gamma g } } \tan ^ { - 1 } \left( \sqrt { \frac { 2 \gamma } { g } } \mathrm {~V} _ { 0 } \right)$