A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to
(1) $x ^ { 3 }$
(2) $e ^ { x }$
(3) $x$
(4) $\log _ { e } x$

A particle which is experiencing a force, given by $\vec { F } = 3 \hat { \mathrm { i } } - 12 \hat { \mathrm { j } }$, undergoes a displacement of $\vec { d } = 4 \hat { \mathrm { i } }$. If the particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement?
(1) 9 J.
(2) 15 J.
(3) 12 J.
(4) 10 J.

A bullet of mass 20 g has an initial speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, just before it starts penetrating a mud wall of thickness 20 cm . If the wall offers a mean resistance of $2.5 \times 10 ^ { - 2 } \mathrm {~N}$, the speed of the bullet after emerging from the other side of the wall is close to:
(1) $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The ball gets deflected by an obstacle on the way. After deflection it moves with $5\%$ of its initial kinetic energy. What is the speed of the ball now?
(1) $19.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $14.41 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $1.00 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle of mass 500 g is moving in a straight line with velocity $v = \mathrm { b } x ^ { \frac { 5 } { 2 } }$. The work done by the net force during its displacement from $x = 0$ to $x = 4 \mathrm {~m}$ is (Take $\mathrm { b } = 0.25 \mathrm {~m} ^ { \frac { - 3 } { 2 } } \mathrm {~s} ^ { - 1 }$).
(1) 2 J
(2) 4 J
(3) 8 J
(4) 16 J

A car accelerates from rest to $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The energy spent in this process is $E \mathrm {~J}$. The energy required to accelerate the car from $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $nE \mathrm {~J}$. The value of $n$ is $\_\_\_\_$.

A particle of mass 10 g moves in a straight line with retardation $2x$, where $x$ is the displacement in SI units. Its loss of kinetic energy for above displacement is $\frac{10^{-n}}{x}\mathrm{~J}$. The value of $n$ will be $\_\_\_\_$.

A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt { x }$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = \mathrm { d }$, will be :
(1) $\frac { m } { 2 \alpha ^ { 2 } d }$
(2) $\frac { \mathrm { md } } { 2 \alpha ^ { 2 } }$
(3) $2 m \alpha ^ { 2 } d$
(4) $\frac { m \alpha ^ { 2 } d } { 2 }$