The potential energy of a $1 \ kg$ particle free to move along the $x-$axis is given by: $U(x) = (\frac{x^4}{4} - \frac{x^2}{2}) \ J$. The total mechanical energy of the particle is $2 \ J$. Then,the maximum speed (in $m/s$) is:

The kinetic energies of two similar cars $A$ and $B$ are $100 \ J$ and $225 \ J$ respectively. On applying brakes,car $A$ stops after $1000 \ m$ and car $B$ stops after $1500 \ m$. If $F_{A}$ and $F_{B}$ are the forces applied by the brakes on cars $A$ and $B$,respectively,then the ratio $F_{A} / F_{B}$ is:

Potential energy of a body of mass $1 \ kg$ free to move along $X$-axis is given by $U(x) = (\frac{x^2}{2} - x) \ J$. If the total mechanical energy of the body is $2 \ J$,then the maximum speed of the body is (Assume only conservative force acts on the body).

$A$ 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 $g=10\,m/s^2$]

$A$ bullet of mass $20 \,g$ leaves a rifle at an initial speed $100 \,m/s$ and strikes a target at the same level with speed $50 \,m/s$. The amount of work done by the resistance of air will be ......... $J$.

$A$ bus moving on a level road with a velocity $v$ can be stopped at a distance of $x$ by the application of a retarding force $F$. The load on the bus is increased by $25\%$ by boarding the passengers. Now,if the bus is moving with the same speed and if the same retarding force is applied,the distance travelled by the bus before it stops is