$300 \, J$ of work is done in sliding up a $2 \, kg$ block on an inclined plane to a height of $10 \, m$. Taking the value of acceleration due to gravity $g$ to be $10 \, m/s^2$,the work done against friction is ........ $J$.

State whether the following statements are true or false:
$(a)$ Power is the vector product of force and velocity.
$(b)$ $1\,kg\,m = 9.8\,J$
$(c)$ When a teacher rubs a blackboard with a duster,they do no work.
$(d)$ Kinetic energy can be negative.

$A$ and $B$ are blocks of same mass $m$ exactly equivalent to each other. Both are placed on a frictionless surface connected by a spring. The natural length of the spring is $L$ and the force constant is $K$. Initially,the spring is at its natural length. Another equivalent block $C$ of mass $m$ travelling at speed $v$ along the line joining $A$ and $B$ collides with $A$. In an ideal condition,the maximum compression of the spring is:

Two balls,having linear momenta $\vec{p}_1 = p \hat{i}$ and $\vec{p}_2 = -p \hat{i}$,undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1^{\prime}$ and $\vec{p}_2^{\prime}$ be their final momenta. Which of the following option$(s)$ is (are) $NOT ALLOWED$ for any non-zero value of $p, a_1, a_2, b_1, b_2, c_1$ and $c_2$?
$(A)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j}$
$(B)$ $\vec{p}_1^{\prime} = c_1 \hat{k}$,$\vec{p}_2^{\prime} = c_2 \hat{k}$
$(C)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j} - c_1 \hat{k}$
$(D)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_1 \hat{j}$

$A$ ball moving with speed $v$ hits another identical ball at rest. The two balls stick together after the collision. If the specific heat of the material of the balls is $S$,the temperature rise resulting from the collision is

$A$ body of mass $4 \ kg$ is moving with a momentum of $8 \ kg \ m/s$. $A$ force of $0.2 \ N$ acts on it in the direction of motion for $10 \ s$. The increase in kinetic energy in joules is: