A train is moving with a velocity of $20\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ on this, dust is falling at the rate of $50\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$min$}\right.$. The extra force required to move this train with a constant velocity will be

The correct option is A

$16.66N$

Step1: Given data and assumptions.

The velocity of the train, $v=20\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Rate of falling dust with respect to time, $\frac{dm}{dt}=50\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$min$}\right.=\frac{50}{60}\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$sec$}\right.$

Step2: Find the extra force required to move the train with constant velocity.

We know that,

$F=\frac{dP}{dt}$

Where, $F$ is force and $\frac{dP}{dt}$ change in momentum with respect to time.

$\Rightarrow$$F=\frac{d\left(mv\right)}{dt}$ $\left[\because P=mv\right]$

$\Rightarrow$$F=v\times \frac{dm}{dt}$

$\Rightarrow$$F=20\times \frac{50}{60}$

$\Rightarrow$$F=16.66N$

Hence, option A is correct.