Work-energy theorem

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 $g=10$ m/s²)

A particle of charge $q$ and mass $m$ is projected from origin with an initial velocity $\vec v=\left(\dfrac{v_0}{\sqrt2}\hat x+\dfrac{v_0}{\sqrt2}\hat y\right)$. There exists a uniform magnetic field $\vec B=B_0\hat z$ and a space varying electric field $\vec E=E_0e^{-\lambda x}\hat x$ within the region $0\le x\le L$. After travelling a distance such that $x$-coordinate has changed from $x=0$ to $x=L$, the change in the kinetic energy is ________.

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 m/s. The work done by the unknown resistive force is ________ J.

(take $g=10$ m/s²)

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

A table tennis ball has radius $(3/2)\times10^{-2}\,\mathrm{m}$ and mass $(22/7)\times10^{-3}\,\mathrm{kg}$. It is slowly pushed down into a swimming pool to a depth of $d = 0.7\,\mathrm{m}$ below the water surface and then released from rest. It emerges from the water surface at speed $v$, without getting wet, and rises up to a height $H$. Which of the following option(s) is(are) correct?

[Given: $\pi = 22/7$, $g = 10\,\mathrm{m\,s^{-2}}$, density of water $= 1\times10^3\,\mathrm{kg\,m^{-3}}$, viscosity of water $= 1\times10^{-3}$ Pa-s.]

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle.

Reason (R) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

A particle experiences a variable force $\vec F=\left(4x\hat i+3y^2\hat j\right)$ 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 :

A 0.5 kg block moving at a speed of $12\ \mathrm{ms^{-1}}$ compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be ______ $\mathrm{Nm^{-1}}$.

A block of mass 2 kg moving on a horizontal surface with speed of 4 ms$^{-1}$ enters a rough surface ranging from $x=0.5$ m to $x=1.5$ m. The retarding force in this range of rough surface is related to distance by $F=-kx$ where $k=12$ Nm$^{-1}$. The speed of the block as it just crosses the rough surface will be :

A particle of mass 500 gm is moving in a straight line with velocity $v=bx^{5/2}$. 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$^{-1}$).

Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :

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

A body of mass 8 kg and another of mass 2 kg are moving with equal kinetic energy. The ratio of their respective momenta will be :

The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at $(4+x)$ cm inside the block. The value of $x$ is :

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

If momentum of a body is increased by 20%, then its kinetic energy increases by

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $dK/dt = \gamma t$, where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A ball is projected from the ground at an angle of $45^\circ$ with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^\circ$ with the horizontal surface. The maximum height it reaches after the bounce, in metres, is ___________.

A body of mass $m=10^{-2}$ kg is moving in a medium and experiences a frictional force $F=-kv^2$. Its initial speed is $v_0=10$ ms$^{-1}$. If, after 10 s, its energy is $\dfrac18mv_0^2$, the value of k will be :

A point particle of mass $m$, moves along the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals μ. The particle is released, from rest, from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR.

The values of the coefficient of friction μ and the distance $x$(=QR), are, respectively close to :