A railroad car of mass M initially rests on frictionless rails. A...

Let the velocity of the man relative to the ground be $v_m$ and the velocity of the car relative to the ground be $v_c = -v$.

Since no external horizontal force acts on the system, the total linear momentum is conserved.

$m v_m + M v_c = 0$

$m v_m - M v = 0$

$v_m = \dfrac{M}{m} v$

The velocity at which the man approaches the engine is his velocity relative to the car.

$v_{rel} = v_m - v_c$

$v_{rel} = \dfrac{M}{m} v - (-v)$

$v_{rel} = \left(1 + \dfrac{M}{m}\right) v$