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) $t^{1/2}$

A body of mass $m$, accelerates uniformly from rest to $v _ { 1 }$ in time $t _ { 1 }$. The instantaneous power delivered to the body as a function of time $t$ is
(1) $\frac { m v _ { 1 } t } { t _ { 1 } }$
(2) $\frac { m v _ { 1 } ^ { 2 } t } { t _ { 1 } ^ { 2 } }$
(3) $\frac { m v _ { 1 } t ^ { 2 } } { t _ { 1 } }$
(4) $\frac { m v _ { 1 } ^ { 2 } t } { t _ { 1 } }$

A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function time is given by
(1) $\frac{mv^2}{T^2} \cdot t$
(2) $\frac{mv^2}{T^2} \cdot t^2$
(3) $\frac{1}{2}\frac{\mathrm{mv}^2}{\mathrm{T}^2} \cdot \mathrm{t}$
(4) $\frac{1}{2}\frac{\mathrm{mv}^2}{\mathrm{T}^2} \cdot \mathrm{t}^2$

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of $3 \mathrm {~m} / \mathrm { s }$. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator ($\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$) must be at least:
(1) 56300 W
(2) 62360 W
(3) 48000 W
(4) 66000 W

An automobile of mass $m$ accelerates starting from the origin and initially at rest, while the engine supplies constant power $P$. The position is given as a function of time by:
(1) $\left( \frac { 9 P } { 8 m } \right) ^ { \frac { 1 } { 2 } } t ^ { \frac { 3 } { 2 } }$
(2) $\left( \frac { 8 P } { 9 m } \right) ^ { \frac { 1 } { 2 } } t ^ { \frac { 2 } { 3 } }$
(3) $\left( \frac { 9 m } { 8 P } \right) ^ { \frac { 1 } { 2 } } t ^ { \frac { 3 } { 2 } }$
(4) $\left( \frac { 8 P } { 9 m } \right) ^ { \frac { 1 } { 2 } } t ^ { \frac { 3 } { 2 } }$

The ratio of powers of two motors is $\frac{3\sqrt{x}}{\sqrt{x+1}}$, that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of $x$ will be
(1) 16
(2) 2
(3) 2.4
(4) 4

A block of mass 5 kg starting from rest pulled up on a smooth incline plane making an angle of $30 ^ { \circ }$ with horizontal with an effective acceleration of $1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The power delivered by the pulling force at $t = 10 \mathrm {~s}$ from the start is $\_\_\_\_$ W. [Use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$] (Calculate the nearest integer value)

If the maximum load carried by an elevator is 1400 kg ( 600 kg -Passengers + 800 kg -elevator) , which is moving up with a uniform speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the frictional force acting on it is 2000 N , then the maximum power used by the motor is $\_\_\_\_$ $\mathrm { kW } . \quad g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

Q22. A string is wrapped around the rim of a wheel of moment of inertia $0.40 \mathrm { kgm } ^ { 2 }$ and radius 10 cm . The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N . The angular velocity of the wheel after 10 s is $x \mathrm { rad } / \mathrm { s }$, where $x$ is $\_\_\_\_$