Two long horizontal smooth rails are 1.0 m apart and joined at one end by a 2.0 Ω resistor. A uniform vertical field B=0.50 T exists everywhere. A…

The magnetic force on the rod is $-B^2l^2v/R$, so $m\,dv/dt=-B^2l^2v/R$ and $v=v_0e^{-t/\tau}$ with $\tau=mR/(B^2l^2)=0.2\times2/0.25=1.6\ \mathrm{s}$. (A) is false: the speed decays exponentially and never reaches zero in finite time; $1.6\ \mathrm{s}$ is only the time constant. (B) is true: $x=\int_0^\infty v\,dt=v_0\tau=5\times1.6=8.0\ \mathrm{m}$. (C) is false: all the kinetic energy becomes heat, $\tfrac12 mv_0^2=\tfrac12(0.2)(25)=2.5\ \mathrm{J}$, not $5\ \mathrm{J}$. (D) is false: the impulse gives $Blq=mv_0$, so $q=0.2\times5/0.5=2.0\ \mathrm{C}$, not $4.0\ \mathrm{C}$. The physical reason is that the induced emf $Blv$ drives a current $Blv/R$ which opposes the motion (Lenz's law), and the opposing force weakens as the rod slows, which is why the decay is exponential rather than uniform.