Two particles,of masses $M$ and $2M$,moving as shown,with speeds of $10\, m/s$ and $5\, m/s$,collide elastically at the origin. After the collision,they move along the indicated directions with speeds $v_1$ and $v_2$ respectively. The values of $v_1$ and $v_2$ are nearly:

$A$ body of mass $5\,kg$ rests on a rough horizontal surface of coefficient of friction $0.2$. The body is pulled through a distance of $10\,m$ by a horizontal force of $25\,N$. The kinetic energy acquired by it is ......... $J$.

$A$ bomb is kept stationary at a point. It suddenly explodes into two fragments of masses $1\,g$ and $3\,g$. The total kinetic energy ($K$.$E$.) of the fragments is $6.4 \times 10^4\,J$. What is the $K$.$E$. of the smaller fragment?

$A$ block of mass $5 \ kg$ is lifted to a height of $5 \ m$ by a force of $60 \ N$. Find:
$(1)$ Work done by the force.
$(2)$ Potential energy of the block at $5 \ m$.
$(3)$ Kinetic energy of the block at $5 \ m$.
$(4)$ Velocity of the block at $5 \ m$. (Take $g = 9.8 \ m/s^2$)

$A$ ball loses $15.0\%$ of its kinetic energy when it bounces back from a concrete wall. With what speed must you throw it vertically down from a height of $12.4\, m$ to have it bounce back to the same height (ignore air resistance)? ............. $m / s$

The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:
$(a)$ Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
$(b)$ Work done by gravitational force in the above case.
$(c)$ Work done by friction on a body sliding down an inclined plane.
$(d)$ Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity.
$(e)$ Work done by the resistive force of air on a vibrating pendulum in bringing it to rest.