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 block of mass 10 kg is moving along $x$-axis under the action of force $F = 5 x \mathrm {~N}$. The work done by the force in moving the block from $x = 2 \mathrm {~m}$ to 4 m will be $\_\_\_\_$ J.

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 force $\vec{F} = (2 + 3x)\hat{i}$ acts on a particle in the $x$ direction where $F$ is in Newton and $x$ is in meter. The work done by this force during a displacement from $x = 0$ to $x = 4$ m is $\_\_\_\_$ J.

The bob of a pendulum was released from a horizontal position. The length of the pendulum is 10 m. If it dissipates $10 \%$ of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$]
(1) $6 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $5 \sqrt { 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $5 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $2 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point $B$ is: (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ).
(1) $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $\sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2 \sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

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) $\frac { m \alpha ^ { 2 } d } { 2 }$

A bob of mass $m$ is suspended by a light string of length $L$. It is imparted a minimum horizontal velocity at the lowest point $A$ such that it just completes half circle reaching the top most position $B$. The ratio of kinetic energies $\frac { ( \text { K.E. } ) _ { A } } { ( \text { K.E. } ) _ { B } }$ is:
(1) $3 : 2$
(2) $5 : 1$
(3) $2 : 5$
(4) $1 : 5$

A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be: Case 1: Vertically upward Case 2: Along the ramp
(1) $1:2$
(2) $\sqrt{3}:2$
(3) $2:1$
(4) $1:1$

If a rubber ball falls from a height $h$ and rebounds upto the height of $h / 2$. The percentage loss of total energy of the initial system as well as velocity of ball before it strikes the ground, respectively, are :
(1) $50 \% , \sqrt { 2 \mathrm { gh } }$
(2) $50 \% , \sqrt { \mathrm { gh } }$
(3) $40 \% , \sqrt { 2 \mathrm { gh } }$
(4) $50 \% , \sqrt { \frac { \mathrm { gh } } { 2 } }$

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h _ { 1 }$ and $h _ { 2 }$, respectively, above the initial level. The ratio $h _ { 1 } : h _ { 2 }$ is $\frac { n } { 10 }$. The value of $n$ is $\_\_\_\_$.

A body of mass $m$ connected to a massless and unstretchable string goes in a vertical circle of radius $R$ under gravity $g$. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $n\sqrt{gR}$, where $n \geq 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is:
(1) $\frac{n^2}{n^2 + 4}$
(2) $\frac{n^2 + 4}{n^2}$
(3) $\frac{n+4}{n}$
(4) $\frac{n}{n+4}$