$A$ ball of mass $1 \ kg$ moves in a straight line with velocity $v = c x^\alpha$, where $c = 1$ ($SI$ unit) and $\alpha$ is a constant. If the work done by the net force during its displacement from $x = 0$ to $x = 4 \ m$ is $128 \ J$, then the value of $\alpha$ is

$A$ closed circular tube of average radius $15\,cm$,whose inner walls are rough,is kept in a vertical plane. $A$ block of mass $1\,kg$ just fits inside the tube. The speed of the block is $22\,m/s$ when it is introduced at the top of the tube. After completing five oscillations,the block stops at the bottom region of the tube. The work done by the tube on the block is $......J$. [Given $g=10\,m/s^2$]

$A$ ball of mass $20 \ kg$ is at rest on a hill of height $100 \ m$. It starts rolling down,reaches the ground,climbs another hill of height $30 \ m$,and finally rolls down to a horizontal platform at a height of $20 \ m$ from the ground. What will be its velocity at this point? (Assume $g = 10 \ m/s^2$)

$A$ body is projected vertically upwards with an initial velocity of $7 \, m/s$. At what height will its kinetic energy become half of its initial value (in $, m$)?

$A$ body of mass $2 \, kg$ slides down a curved track which is a quadrant of a circle of radius $1 \, m$ as shown in the figure. All the surfaces are frictionless. If the body starts from rest,its speed at the bottom of the track is ............. $m/s$.

$A$ particle is released from a height $H$. At a certain height,its kinetic energy is half of its potential energy with reference to the surface of the earth. The height and speed of the particle at that instant are respectively: