Work, Energy and Power

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

A 1 kg block subjected to two simultaneous forces $(2\hat{i}+3\hat{j}+4\hat{k})$ N and $(3\hat{i}-\hat{j}-2\hat{k})$ N is moved a distance of 25 m along $(3\hat{i}-4\hat{j})$ direction. The work done in this process is ______ J.

A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F}=(2t\hat{i}+6t^2\hat{j})$ N, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$-axis respectively. The power produced by the force at $t=2$ s is ______ W.

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 m/s. The work done by the frictional force in time 2 s is ______ J. (Take $g=10\ \mathrm{m/s^2}$)

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 m/s. The work done by the unknown resistive force is ________ J. (take $g=10\ \mathrm{m/s^2}$)

A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height $h$ as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height $h=\alpha R$. The value of $\alpha$ is ________.

A force $\vec{F} = 2\hat{i} + b\hat{j} + \hat{k}$ 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 force $F=\alpha+\beta x^2$ 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 $\alpha=1$ N then $\beta$ will be

A body of mass '$m$' connected to a massless and unstretchable string goes in verticle 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, $n \ge 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities $v_A = 5$ m/s, $v_B = 2$ m/s, $v_C = 4$ m/s. If we wait sufficiently long for elastic collision to happen, then $v_A = 4$ m/s, $v_B = 2$ m/s, $v_C = 5$ m/s will be the final velocities. Reason (R): In an elastic collision between identical masses, two objects exchange their velocities. In the light of the above statements, choose the correct answer from the options given below :