I am confused with the way we need to use different approach in some similar questions from Kleppner and Kolenkow mechanics like those given below:


> 1. An empty freight car of mass M starts from rest under an applied
force F. At the same time, sand begins to run into the car at steady
rate b from a hopper at rest along the track.
Find the speed when a mass of sand m has been transferred.

 


$P(0) =0$

$ P(t) = (M + bt)v
$

$impulse = $$\int
 P = (M + bt)v =
$$\int 
Fdt  = F
$$\int
dt = Ft$

$ v =
Ft/
(M + bt)$.................................(1)

But I got the following result which as easily seen comes from the integration of

 $1/[(M+bt)/M]*dt$,

$v=F/b*ln[(M+ bt)/M] $................(2)

On the other hand we get the same result given by eq(2)  for the following question:
  
> 2. A freight car of mass M contains a mass of sand m. At t = 0 a
constant horizontal force F is applied in the direction of rolling
and at the same time a port in the bottom is opened to let the sand
flow out at constant rate dm/dt. Find the speed of the freight car
when all the sand is gone. Assume the freight car is at rest at t = 0.


I couldn't really understand why we use different approaches to deal with these two question.(Actually,the only difference is in the last step i.e. while solving the integral). But why the twist all of a sudden?

 I would be very thankful if someone could help me understanding the difference between these two question, maybe the difference in the physical context these two questions hold that can justify why we solve them differently. 

PS: I am not asking for anyone to solve them cause I know the solution.It's just  that I wonder if I've missed out certain underlying physics which creates the difference.