The figure shows an experimental plot for discharging of a capacitor in an R-C circuit. The time constant $$\tau $$ of this circuit lies between

A particle of mass $$m$$ is at rest at the origin at time $$t=0.$$ It is subjected to a force $$F\left( t \right) = {F_0}{e^{ - bt}}$$ in the $$x$$ direction. Its speed $$v(t)$$ is depicted by which of the following curves?

A charge $$Q$$ is uniformly distributed over the surface of non-conducting disc of radius $$R.$$ The disc rotates about an axis perpendicular to its plane and passing through its center with an angular velocity $$\omega .$$ As a result of this rotation a magnetic field of induction $$B$$ is obtained at the center of the disc. If we keep both the amount of charge placed on the disc and its angular velocity to be constant and very the radius of the disc then the variation of the magnetic induction at the center of the disc will be represented by the figure :

The mass of a spaceship is $$1000$$ $$kg.$$ It is to be launched from the earth's surface out into free space. The value of $$g$$ and $$R$$ (radius of earth ) are $$10\,m/{s^2}$$ and $$6400$$ $$km$$ respectively. The required energy for this work will be: