If $t _ { 1 }$ and $t _ { 2 }$ are the times of flight of two particles having the same initial velocity $u$ and range R on the horizontal, then $t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 }$ is equal to
(1) $\frac { u ^ { 2 } } { g }$
(2) $\frac { 4 u ^ { 2 } } { g ^ { 2 } }$
(3) $\frac { u ^ { 2 } } { 2 g }$
(4) 1

A body of mass 5 kg under the action of constant force $\vec{F} = F_x \hat{i} + F_y \hat{j}$ has velocity at $\mathrm{t} = 0 \mathrm{~s}$ as $\overrightarrow{\mathrm{v}} = (6\hat{\mathrm{i}} - 2\hat{\mathrm{j}}) \mathrm{m/s}$ and at $\mathrm{t} = 10 \mathrm{~s}$ as $\overrightarrow{\mathrm{v}} = +6\hat{\mathrm{j}} \mathrm{m/s}$. The force $\overrightarrow{\mathrm{F}}$ is:
(1) $(-3\hat{\mathrm{i}} + 4\hat{\mathrm{j}}) \mathrm{N}$
(2) $\left(-\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}\right) \mathrm{N}$
(3) $(3\hat{\mathrm{i}} - 4\hat{\mathrm{j}}) \mathrm{N}$
(4) $\left(\frac{3}{5}\hat{\mathrm{i}} - \frac{4}{5}\hat{\mathrm{j}}\right) \mathrm{N}$

A particle of mass m is moving in a straight line with momentum p. Starting at time $t = 0$, a force $F = k \mathrm { t }$ acts in the same direction on the moving particle during time interval T so that its momentum changes from p to 3p. Here $k$ is a constant. The value of T is
(1) $2\sqrt { \frac { k } { p } }$
(2) $2\sqrt { \frac { \mathrm { p } } { k } }$
(3) $\sqrt { \frac { 2k } { p } }$
(4) $\sqrt { \frac { 2p } { \mathrm { k } } }$

A body of mass 500 g moves along $x$-axis such that it's velocity varies with displacement $x$ according to the relation $v = 10 \sqrt { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the force acting on the body is:
(1) 125 N
(2) 25 N
(3) 166 N
(4) 5 N