An object is projected with kinetic energy $K$ from a point $A$ at an angle $60^{\circ}$ with the horizontal. The ratio of the difference in kinetic energies at points $B$ and $C$ to that at point $A$ (see figure), in the absence of air friction is :

Two blocks with masses 100 g and 200 g are attached to the ends of springs $A$ and $B$ as shown in figure. The energy stored in $A$ is $E$. The energy stored in $B$, when spring constants $k_A, k_B$ of $A$ and $B$, respectively satisfy the relation $4 k_A=3 k_B$, is :

Given below are two statements :

Statement I : An object moves from position $r_1$ to position $r_2$ under a conservative force field $\vec{F}$. The work done by the force is $W=-\int\limits_{r_1}^{r_2} \vec{F} \cdot \overrightarrow{d r}$.

Statement II : Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.

In the light of the above statements, choose the correct answer from the options given below :

A body of mass 2 kg is moving along x-direction such that its displacement as function of time is given by $x(t) = \alpha t^2 + \beta t + \gamma$ m, where $\alpha = 1 \; m/s^2$, $\beta = 1 \; m/s$ and $\gamma = 1 \; m$. The work done on the body during the time interval $t = 2 \; s$ to $t = 3 \; s$, is ________ J.

Potential energy ( $V$ ) versus distance ( $x$ ) is given by the graph. Rank various regions as per the magnitudes of the force ( $F$ ) acting on a particle from high to low.

A body of mass 1 kg moves along a straight line with a velocity $v=2 x^2$. The work done by the body during displacement from $x=0$ to 5 m is $\_\_\_\_$ J.

The velocity at which 6 kg mass (shown in figure) strikes the ground when it is released from a height of 6 m above the ground is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$. Assume pulley is massless and string is light and inextensible. (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

The rain drop of mass 1 g , starts with zero velocity from a height of 1 km . It hits the ground with a speed of $5 \mathrm{~m} / \mathrm{s}$. The work done by the unknown resistive force is $\_\_\_\_$ J.

(take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

A mass of 1 kg is kept on a inclined plane with $30^{\circ}$ inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of $4 \mathrm{~m} / \mathrm{s}$. The work done by the frictional force in time 2 s is $\_\_\_\_$ J. (Take $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A spherical ball of mass 2 kg falls from a height of 10 m and is brought to rest after penetrating 10 cm into sand.

The average force exerted by sand on the ball is _________ N.

(Take $g = 10 \ \mathrm{m/s^2}$)

Which one of the following forces cannot be expressed in terms of potential energy?

An object of mass 1000 g experiences a time dependent force $\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N$. The power generated by the force at time $t$ is:

A block of mass 25 kg is pulled along a horizontal surface by a force at an angle $45^{\circ}$ with the horizontal. The friction coefficient between the block and the surface is 0.25 . The block travels at a uniform velocity. The work done by the applied force during a displacement of 5 m of the block is :

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., $\frac{dm}{dt} \propto \sqrt{v}$. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

A bead of mass ' $m$ ' slides without friction on the wall of a vertical circular hoop of radius ' $R$ ' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is ' $R$ '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' $R$ ', would be (spring constant is ' $k$ ', $g$ is accleration due to gravity)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason $\mathbf{R}$

Assertion A: In a central force field, the work done is independent of the path chosen.

Reason R: Every force encountered in mechanics does not have an associated potential energy.

In the light of the above statements, choose the most appropriate answer from the options given below

A force $\mathrm{F}=\alpha+\beta \mathrm{x}^2$ acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m . If the constant $\alpha=1 \mathrm{~N}$ then $\beta$ will be

A ball having kinetic energy KE, is projected at an angle of $60^{\circ}$ from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?

A force $\overrightarrow{\mathrm{F}}=2 \hat{i}+\mathrm{b} \hat{j}+\hat{k}$ is applied on a particle and it undergoes a displacement $\hat{i}-2 \hat{j}-\hat{k}$ What will be the value of $b$, if work done on the particle is zero.

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 :

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :

Three bodies A, B and C have equal kinetic energies and their masses are $400 \mathrm{~g}, 1.2 \mathrm{~kg}$ and $1.6 \mathrm{~kg}$ respectively. The ratio of their linear momenta is :

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

A bullet of mass $50 \mathrm{~g}$ is fired with a speed $100 \mathrm{~m} / \mathrm{s}$ on a plywood and emerges with $40 \mathrm{~m} / \mathrm{s}$. The percentage loss of kinetic energy is :

Four particles $A, B, C, D$ of mass $\frac{m}{2}, m, 2 m, 4 m$, have same momentum, respectively. The particle with maximum kinetic energy is :

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

A body of mass $50 \mathrm{~kg}$ is lifted to a height of $20 \mathrm{~m}$ 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 :

A body of $m \mathrm{~kg}$ slides from rest along the curve of vertical circle from point $A$ to $B$ in friction less path. The velocity of the body at $B$ is:

(given, $R=14 \mathrm{~m}, g=10 \mathrm{~m} / \mathrm{s}^2$ and $\sqrt{2}=1.4$)

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 ball before it strikes the ground, respectively, are :

A body of mass $2 \mathrm{~kg}$ begins to move under the action of a time dependent force given by $\vec{F}=\left(6 t \hat{i}+6 t^2 \hat{j}\right) N$. The power developed by the force at the time $t$ is given by:

A block of mass $1 \mathrm{~kg}$ is pushed up a surface inclined to horizontal at an angle of $60^{\circ}$ by a force of $10 \mathrm{~N}$ parallel to the inclined surface as shown in figure. When the block is pushed up by $10 \mathrm{~m}$ along inclined surface, the work done against frictional force is :

$\left[g=10 \mathrm{~m} / \mathrm{s}^2\right]$

A particle is placed at the point $A$ of a frictionless track $A B C$ as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point B is :

(Take $g=10 \mathrm{~m} / \mathrm{s}^2$).

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 $\frac{(K . E)_A}{(K . E)_B}$ is :

A block of mass $100 \mathrm{~kg}$ slides over a distance of $10 \mathrm{~m}$ on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction $(\operatorname{in} J$) is :

The potential energy function (in $J$ ) of a particle in a region of space is given as $U=\left(2 x^2+3 y^3+2 z\right)$. Here $x, y$ and $z$ are in meter. The magnitude of $x$-component of force (in $N$ ) acting on the particle at point $P(1,2,3) \mathrm{m}$ is :

A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle $(\theta)$ of thread deflection in the extreme position will be :

A bullet is fired into a fixed target looses one third of its velocity after travelling $4 \mathrm{~cm}$. It penetrates further $\mathrm{D} \times 10^{-3} \mathrm{~m}$ before coming to rest. The value of $\mathrm{D}$ is :

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

Two bodies are having kinetic energies in the ratio 16 : 9. If they have same linear momentum, the ratio of their masses respectively is :

Given below are two statements:

Statement I : A truck and a car moving with same kinetic energy are brought to rest by applying breaks which provide equal retarding forces. Both come to rest in equal distance.

Statement II : A car moving towards east takes a turn and moves towards north, the speed remains unchanged. The acceleration of the car is zero.

In the light of given statements, choose the most appropriate answer from the options given below

A bullet of mass $0.1 \mathrm{~kg}$ moving horizontally with speed $400 \mathrm{~ms}^{-1}$ hits a wooden block of mass $3.9 \mathrm{~kg}$ kept on a horizontal rough surface. The bullet gets embedded into the block and moves $20 \mathrm{~m}$ before coming to rest. The coefficient of friction between the block and the surface is __________.

(Given $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Identify the correct statements from the following :

A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.

B. Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.

C. Work done by friction on a body sliding down an inclined plane is positive.

D. Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero.

E. Work done by the air resistance on an oscillating pendulum is negative.

Choose the correct answer from the options given below :

A stone is projected at angle $30^{\circ}$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be -

A ball is projected with kinetic energy E, at an angle of $60^{\circ}$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :

A bullet of mass $200 \mathrm{~g}$ having initial kinetic energy $90 \mathrm{~J}$ is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to $40 \mathrm{~J}$ within $1 \mathrm{~s}$, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

Sand is being dropped from a stationary dropper at a rate of $0.5 \,\mathrm{kgs}^{-1}$ on a conveyor belt moving with a velocity of $5 \mathrm{~ms}^{-1}$. The power needed to keep the belt moving with the same velocity will be :

As per the given figure, two blocks each of mass $250 \mathrm{~g}$ are connected to a spring of spring constant $2 \,\mathrm{Nm}^{-1}$. If both are given velocity $v$ in opposite directions, then maximum elongation of the spring is :

A bag of sand of mass 9.8 kg is suspended by a rope. A bullet of 200 g travelling with speed 10 ms^$-$1 gets embedded in it, then loss of kinetic energy will be :

A body of mass $0.5 \mathrm{~kg}$ travels on straight line path with velocity $v=\left(3 x^{2}+4\right) \mathrm{m} / \mathrm{s}$. The net workdone by the force during its displacement from $x=0$ to $x=2 \mathrm{~m}$ is :

In the given figure, the block of mass m is dropped from the point 'A'. The expression for kinetic energy of block when it reaches point 'B' is

A particle of mass 500 gm is moving in a straight line with velocity v = b x^5/2. The work done by the net force during its displacement from x = 0 to x = 4 m is : (Take b = 0.25 m^$-$3/2 s^$-$1).

Arrange the four graphs in descending order of total work done; where W₁, W₂, W₃ and W₄ are the work done corresponding to figure a, b, c and d respectively.

A particle experiences a variable force $\overrightarrow F = \left( {4x\widehat i + 3{y^2}\widehat j} \right)$ in a horizontal x-y plane. Assume distance in meters and force is newton. If the particle moves from point (1, 2) to point (2, 3) in the x-y plane, then Kinetic Energy changes by :

The value of the coefficient of friction $\mu $ and the distance $x$ $(=QR),$ are, respectively close to:

If two springs ${S_1}$ and ${S_2}$ of force constants ${k_1}$ and ${k_2}$, respectively, are stretched by the same force, it is found that more work is done on spring ${S_1}$ than on spring ${S_2}$.

STATEMENT 1: If stretched by the same amount work done on ${S_1}$, Work done on ${S_1}$ is more than ${S_2}$
STATEMENT 2: ${k_1} < {k_2}$

The total mechanical energy of the particle is $2J.$ Then, the maximum speed (in $m/s$) is