$A$ 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 $\frac{1}{8} mv_0^2$,the value of $k$ will be

Place a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \ cm$ and the right one at $90.00 \ cm$. When you attempt to move both fingers slowly towards the center,initially only the left finger slips with respect to the scale and the right finger does not. After some distance,the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center $(50.00 \ cm)$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are $0.40$ and $0.32$,respectively,the value of $x_R$ (in $cm$) is:

$A$ $100 \ kg$ gun fires a ball of $1 \ kg$ horizontally from a cliff of height $500 \ m$. It falls on the ground at a distance of $400 \ m$ from the bottom of the cliff. The recoil velocity of the gun is (Acceleration due to gravity $= 10 \ ms^{-2}$). (in $ms^{-1}$)

$A$ $0.5\, kg$ ball moving with a speed of $12\, m/s$ strikes a hard wall at an angle of $30^\circ$ with the wall. It is reflected with the same speed at the same angle. If the ball is in contact with the wall for $0.25\, s$,the average force acting on the wall is ........... $N$.

In which of the following graphs,the total change in momentum is zero?

One end of a spring of force constant $150 \text{ dyne cm}^{-1}$ is connected to a block of mass $0.2 \text{ kg}$ kept on a rough horizontal surface of coefficient of friction $0.3$. The other end of the spring is connected to a rigid support as shown in the figure and the spring is initially undeformed. The maximum velocity $v$ that can be given to the block so that it travels only in one direction is . . . . . . $\text{ms}^{-1}$. (Acceleration due to gravity $= 10 \text{ ms}^{-2}$)