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A body is rolling a \( 0.5 \, \text{kg} \) ball on the frictionless floor with the speed of \( 20 \, \text{ms}^{-1} \). 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?

In a spring gun having spring constant \( 100 \, \text{N/m} \) a small ball ‘B’ of mass \( 100 \, \text{g} \) is put in its barrel (as shown in figure) by compressing the spring through \( 0.05 \, \text{m} \). There should be a box placed at a distance '\( d \)' on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of \( 2 \, \text{m} \) above the ground. The value of \( d \) is __________ \( \text{m} \). \( ( g = 10 \, \text{m/s}^{2} ) \).

A circular hole of radius \( \left(\frac{a}{2}\right) \) is cut out of a circular disc of radius ‘\( a \)’ as shown in figure. The centroid of the remaining circular portion with respect to point ‘\( O \)’ will be:

A large block of wood of mass \( M = 5.99 \, \text{kg} \) is hanging from two long massless cords. A bullet of mass \( m = 10 \, \text{g} \) is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance \( h = 9.8 \, \text{cm} \) before the (block + bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is: (Take \( g = 9.8 \, \text{m s}^{-2} \)).

A uniform chain of length \( 3 \, \text{meter} \) and mass \( 3 \, \text{kg} \) overhangs a smooth table with \( 2 \, \text{meter} \) laying on the table. If \( k \) is the kinetic energy of the chain in joule as it completely slips off the table, then the value of \( k \) is ________. (Take \( g = 10 \, \text{m/s}^{2} \))

A body of mass \( 2 \, \text{kg} \) moving with a speed of \( 4 \, \text{m/s} \), makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is \( \dfrac{x}{10} \, \text{m/s} \). Then the value of \( x \) is ________________ .

Two billiard of equal mass \(30 g\) strike a rigid wall with same speed of \(108\,\text{ kmph}\) (as shown) but at different angles. if the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball 'a' and ball ‘b’ by the wall along 'X' direction is : 

A ball of mass \( 10 \, \text{kg} \) moving with a velocity \( 10\sqrt{3} \, \text{m}\!/\!\text{s} \) along the x-axis, hits another ball of mass \( 20 \, \text{kg} \) which is at rest. After this collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along y-axis with a speed of \( 10 \, \text{m}\!/\!\text{s} \). The second piece starts moving at an angle of \( 30^\circ \) with respect to the x-axis. The velocity of the ball moving at \( 30^\circ \) with x-axis is \( x \, \text{m}\!/\!\text{s} \). The configuration of pieces after collision is shown in the figure below. The value of \( x \) to the nearest integer is ____________.

These objects \( A \), \( B \) and \( C \) are kept in a straight line on a frictionless horizontal surface. The masses of \( A \), \( B \) and \( C \) are \( m \), \( 2m \) and \( 2m \) respectively. \( A \) moves towards \( B \) with a speed of \( 9 \, \text{m}/\text{s} \) and makes an elastic collision with it. Thereafter \( B \) makes completely inelastic collision with \( C \). All motions occurs along same straight line. The final speed of \( C \) is:

A force of \( {F} = (5y + 20)\hat{j} \, \text{N} \) acts on a particle. The work done by this force when the particle is moved from \( y = 0 \, \text{m} \) to \( y = 10 \, \text{m} \) is ______ \( \text{J} \).

A ball of mass \( 4 \, \text{kg} \), moving with a velocity of \( 10 \, \text{m s}^{-1} \), collides with a spring of length \( 8 \, \text{m} \) and force constant \( 100 \, \text{N m}^{-1} \). The length of the compressed spring is \( x \, \text{m} \). The value of \( x \), to the nearest integer is:

A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time \( t \) is proportional to:

Two solids A and B of mass \( 1 \, \text{kg} \) and \( 2 \, \text{kg} \) respectively are moving with equal linear momentum. The ratio of their kinetic energies \( \text{(K.E.)}_A : \text{(K.E.)}_B \) will be \( \frac{A}{1} \), so the value of A will be ______.

A particle of mass \( m \) moves in a circular orbit under the central potential field, \( U(r) = -\dfrac{C}{r} \), where \( C \) is a positive constant. The correct radius - velocity graph of the particle’s motion is:

Two person A and B perform same amount of work in moving a body through a certain distance \( d \) with application of forces acting at angle \( 45^\circ \) and \( 60^\circ \) with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person B is \( \dfrac{1}{\sqrt{x}} \). The value of \( x \) is ______.

A bullet of \( 4 \, \text{g} \) mass is fired from a gun of mass \( 4 \, \text{kg} \). If the bullet moves with the muzzle speed of \( 50 \, \text{m s}^{-1} \), the impulse imparted to the gun and velocity of recoil of gun are:

A porter lifts a heavy suitcase of mass \( 80 \, \text{kg} \) and at the destination lowers it down by a distance of \( 80 \, \text{cm} \) with a constant velocity. Calculate the workdone by the porter in lowering the suitcase. (Take \( g = 9.8 \, \text{m s}^{-2} \) )

A block moving horizontally on a smooth surface with a speed of \( 40 \,\text{m/s} \) splits into two parts with masses in the ratio of \( 1:2 \). If the smaller part moves at \( 60 \,\text{m/s} \) in the same direction, then the fractional change in kinetic energy is:

A bullet of \( 10 \, \text{g} \), moving with velocity \( v \), collided head-on with the stationary bob of a pendulum and recoils with velocity \( 100 \, \text{m/s} \). The length of the pendulum is \( 0.5 \, \text{m} \) and mass of the bob is \( 1 \, \text{kg} \). The minimum value of \( v = \) ______ \( \text{m/s} \) so that the pendulum describes a circle.
(Assume the string to be inextensible and \( g = 10 \, \text{m/s}^{2} \))

A body at rest is moved along a horizontal straight line by a machine delivering a constant power. The distance moved by the body in time ‘\( t \)’ is proportional to:

An object mass of \( M_1 \) collides with another object of mass \( M_2 \), which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses \( M_2 : M_1 \) is:

A bullet of mass \( 0.1 \, \text{kg} \) is fired on a wooden block to pierce through it, but it stops after moving a distance of \( 50 \, \text{cm} \) into it. If the velocity of bullet before hitting the wood is \( 10 \, \text{m/s} \) and it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is \( 'x' \, \text{N} \).
The value of \( 'x' \) to the nearest integer is __________ .

Two identical blocks \( A \) and \( B \) each of mass \( m \) resting on the smooth horizontal floor are connected by a light spring of natural length \( L \) and spring constant \( K \). A third block \( C \) of mass \( m \) moving with a speed \( v \) along the line joining \( A \) and \( B \) collides with \( A \). The maximum compression in the spring is:

A body of mass \( M \) moving at speed \( V_{0} \) collides elastically with a mass ‘\( m \)’ at rest. After the collision, the two masses move at angles \( \theta_{1} \) and \( \theta_{2} \) with respect to the initial direction of motion of the body of mass \( M \). The largest possible value of the ratio \( \dfrac{M}{m} \), for which the angles \( \theta_{1} \) and \( \theta_{2} \) will be equal, is:

The position of the centre of mass of a uniform semi-circular wire of radius ‘\( R \)’ placed in x–y plane with its centre at the origin and the line joining its ends as x-axis is given by \( \left( 0, \dfrac{xR}{\pi} \right) \). Then, the value of \( |x| \) is ______.

The disc of mass \( M \) with uniform surface mass density \( \sigma \) is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position \( \displaystyle \frac{x}{3} \frac{a}{\pi}, \frac{x}{3} \frac{a}{\pi} \) where \( x \) is _______. (Round off to the Nearest Integer) [\( a \) is an area as shown in the figure]

A block moving horizontally on a smooth surface with a speed of \( 40 \, \text{m s}^{-1} \) splits into two equal parts. If one of the parts moves at \( 60 \, \text{m s}^{-1} \) in the same direction, then the fractional change in the kinetic energy will be \( x : 4 \) where \( x = \) ________.

A rod of mass \(M\) and length \(L\) is lying on a horizontal frictionless surface. A particle of mass ‘\(m\)’ travelling along the surface hits at one end of the rod with a velocity ‘\(u\)’ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses \(\left(\dfrac{m}{M}\right)\) is \(\dfrac{1}{x}\). The value of ‘\(x\)’ will be _____________.

As shown in the figure, a particle of mass \( 10 \, \text{kg} \) is placed at a point A. When the particle is slightly displaced to its right, it starts moving and reaches the point B. The speed of the particle at B is \( x \, \text{m}\!/\!\text{s} \). (Take \( g = 10 \, \text{m}\,/\text{s}^{2} \)) The value of \( x \) to the nearest integer is _____________.