A rod of mass `m` and resistance `r` is placed on fixed, resistanceless, smooth conducting rails (closed by a resistance `R`) and it is projected with an initial velocity `u`.Find its velocity as a function of time.

Explanation

Let at an instant the velocity of the rod be `v`.The `emf` induced in the rod will be `vBl`.The electrically equivalent circuit is shown in the following diagram.
`therefore` Current in the circuit `i=(Blv)/(R+r)`
At time `t`
Magnetic force acting on the rod is `F=ilB` opposite to the motion of the rod.
`ilB=-m(dV)/(dt)`..(1)
`i=(ilB)/(R+r)` ..(2)
Now solving these two equation
`(B^(2)l^(2)V)/(R+r)=-m.(dV)/(dt)`
`-(B^(2)l^(2))/((R+r)m).dt=(dv)/V` let `(B^(2)l^(2))/((R+r)m)=k`
`-K.dt=(dV)/V`
`underset(u)overset(v)int(dV)/V= underset(0)overset(t)int-K.dt`
`In(v/u)=-Kt`
`V=ue^(-kt)`