The co-ordinates of a moving particle at any time '$t$' are given by $x = \alpha t^{3}$ and $y = \beta t^{3}$. The speed of the particle at time '$t$' is given by (1) $3t\sqrt{\alpha^{2} + \beta^{2}}$ (2) $3t^{2}\sqrt{\alpha^{2} + \beta^{2}}$ (3) $t^{2}\sqrt{\alpha^{2} + \beta^{2}}$ (4) $\sqrt{\alpha^{2} + \beta^{2}}$
A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration $f$ and in the second part with constant retardation $r$. The value of $t$ is given by (1) $\sqrt{2s\left(\frac{1}{f} + \frac{1}{r}\right)}$ (2) $2s\left(\frac{1}{f} + \frac{1}{r}\right)$ (3) $\frac{2s}{\frac{1}{f} + \frac{1}{r}}$ (4) $\sqrt{2s(f + r)}$
Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\overrightarrow{\mathrm{u}}$ and the other from rest with uniform acceleration $\overrightarrow{\mathrm{f}}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time. (1) $\frac{u\cos\alpha}{f}$ (2) $\frac{u\sin\alpha}{f}$ (3) $\frac{f\cos\alpha}{u}$ (4) $u\sin\alpha$
A boy playing on the roof of a 10 m high building throws a ball with a speed of $10 \mathrm{~m/s}$ at an angle of $30^{\circ}$ with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground? $$\left[\mathrm{g} = 10 \mathrm{~m/s}^{2}, \sin 30^{\circ} = \frac{1}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}\right]$$ (1) 5.20 m (2) 4.33 m (3) 2.60 m (4) 8.66 m
Two stones are projected from the top of a cliff $h$ metres high, with the same speed $u$, so as to hit the ground at the same spot. If one of the stones is projected at an angle $\theta$ to the horizontal then the $\theta$ equals (1) $u\sqrt{\frac{2}{gh}}$ (2) $\sqrt{\frac{2u}{gh}}$ (3) $2g\sqrt{\frac{u}{h}}$ (4) $2h\sqrt{\frac{u}{g}}$
A block of mass $M$ is pulled along a horizontal frictionless surface by a rope of mass m. If a force P is applied at the free end of the rope, the force exerted by the rope on the block is (1) $\frac{\mathrm{Pm}}{\mathrm{M} + \mathrm{m}}$ (2) $\frac{\mathrm{Pm}}{\mathrm{M} - \mathrm{m}}$ (3) $P$ (4) $\frac{\mathrm{PM}}{\mathrm{M} + \mathrm{m}}$
A spring of spring constant $5 \times 10^{3} \mathrm{~N/m}$ is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is (1) $12.50 \mathrm{~N-m}$ (2) $18.75 \mathrm{~N-m}$ (3) $25.00 \mathrm{~N-m}$ (4) $6.25 \mathrm{~N-m}$
A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time '$t$' is proportional to (1) $t^{3/4}$ (2) $t^{3/2}$ (3) $t^{1/4}$ (4) $t^{1/2}$