A spring-mass system vibrates such that mass travel on surface of coefficient of friction  $\mu$. Mass is released after compressing the spring by distance a and it travels upto  distance $b$ after its equilibrium position, then travelling from $x = -a$ to $x = b$ the reduction in its amplitude will be :-

Under the action of a force, a $2 \,kg$ body moves such that its position $x$ as a function of time $t$ is given by $x=\frac{t^2}{3}$, where $x$ is in metres and $t$ in seconds. The workdone by the force in first two seconds is .......... $J$

The position $x$ of a particle moving along $x$-axis at time $(t)$ is given by the equation $t=\sqrt{x}+2$, where $x$ is in metres and $t$ in seconds. the work done by the force in first four seconds is .............. $J$

$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where

On complete combustion a litre of petrol gives off heat equivalent to $3\times 10^7\,J$. In a test drive, a car weighing $1200\,kg$ including the mass of driver, runs $15\,km$ per litre while moving with a uniform speed on a straight track. Assuming that friction offered by the road surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were  $0.5$.

A raindrop of mass $1.00\, g$ falling from a height of $1\,km$ hits the ground with a speed of $50\,m s^{-1}$. Calculate

$(a)$ the loss of $PE$ of the drop

$(b)$ the gain in $KE$ of the drop

$(c)$ Is the gain in $KE$ equal to loss of $PE$ ? If not why ?

Take, $g = 10\, m s^{-2}$.