The relation between time '$$t$$' and distance '$$x$$' is $$t=\alpha x^2+\beta x$$, where $$\alpha$$ and $$\beta$$ are constants. The relation between acceleration $$(a)$$ and velocity $$(v)$$ is :

An artillery piece of mass $$M_1$$ fires a shell of mass $$M_2$$ horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

The parameter that remains the same for molecules of all gases at a given temperature is :

A rigid wire consists of a semicircular portion of radius $$R$$ and two straight sections. The wire is partially immerged in a perpendicular magnetic field $$B=B_0 \hat{k}$$ as shown in figure. The magnetic force on the wire if it has a current $$i$$ is: