$A$ person trying to lose weight (dieter) lifts a $10\; kg$ mass,one thousand times,to a height of $0.5\; m$ each time. Assume that the potential energy lost each time she lowers the mass is dissipated.
$(a)$ How much work does she do against the gravitational force?
$(b)$ Fat supplies $3.8 \times 10^{7} \;J$ of energy per kilogram which is converted to mechanical energy with a $20\%$ efficiency rate. How much fat will the dieter use up?

$A$ mass of $10 \, g$ moving with a velocity of $100 \, cm/s$ strikes a pendulum bob of mass $10 \, g$. The two masses stick together. The maximum height reached by the system now is ............... $cm$ $(g = 10 \, m/s^2)$.

$A$ ball of mass $2m$ is moving with velocity $v$ on a smooth surface and collides elastically head-on with another ball of mass $m$ which is at rest. If the ball of mass $m$ reaches up to the top of a frictionless elevated plane of height $h$,then the velocity $v$ of the heavy ball must be

$A$ bullet of mass $0.012\;kg$ and horizontal speed $70\;m\;s^{-1}$ strikes a block of wood of mass $0.4\;kg$ and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also,estimate the amount of heat produced in the block.

$A$ particle is projected vertically upwards with a speed of $16 \ m/s$. After some time,when it again passes through the point of projection,its speed is found to be $8 \ m/s$. It is known that the work done by air resistance is the same during upward and downward motion. Then the maximum height attained by the particle is ...................... $m$ $(g = 10 \ m/s^2)$.

$A$ slide with a frictionless curved surface,which becomes horizontal at its lower end,is fixed on the terrace of a building of height $3h$ from the ground,as shown in the figure. $A$ spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec{u}_0 = u_0 \hat{x}$ and falls on the ground at a distance $d$ from the building,making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $e = 1 / \sqrt{3}$. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\vec{u}_0 = \sqrt{2gh} \hat{x}$
$(B)$ $\vec{v} = \sqrt{2gh} \hat{x} + \sqrt{2gh} \hat{z}$
$(C)$ $\theta = 60^{\circ}$
$(D)$ $d / h_1 = 2\sqrt{3}$