How Does Sand Transfer Affect the Speed of a Freight Car?

  • Thread starter geoffrey159
  • Start date
  • Tags
    Locomotive
  • #1
geoffrey159
535
72

Homework Statement


(Kleppner & Kolenkow - Introduction to Mechanics - 3.12)
A sand-spraying locomotive sprays sand horizontally into a freight car situated ahead of it. The locomotive and freight car are not attached. The engineer in the locomotive maintains his speed so that the distance to the freight car is constant. The sand is transferred at a rate dm/dt = 10 kg/s with a velocity of 5 m/s relative to the locomotive. The car starts from rest with an initial mass of 2000 kg. Find its speed after 100 s.

Homework Equations


Conservation of momentum

The Attempt at a Solution



Let ##x_l## be the horizontal position of the locomotive, ##x_f## the horizontal position of the freight, and ##x_s## the horizontal position of a piece of sand about to fall into the freight.
We have the relationships:
1 - ## x_f - x_l = cst ## so that the locomotive and the freight have the same speed: ##v_l = v_f##
2 - ## x_{s} = x_l + x_{s/l} \Rightarrow v_{s} = v_l + v_{s/l} = v_f + 5 ## meters per second.

B - Momentum change
Call ##m(t)## the changing mass of the freight, and ##\triangle m## a small amount of sand falling into the freight in ##\triangle t## seconds.
##P(t) = m(t) v_f(t) + \triangle m v_s(t) ##
##P(t+\triangle t) = (m(t) + \triangle m) v_f(t+\triangle t)##

Since there are no external forces:
## 0 = \frac{dP}{dt} = m(t) \frac{dv_f}{dt} + \frac{dm}{dt} (v_f - v_s) = m(t) \frac{dv_f}{dt} - 5 \frac{dm}{dt} ##

Using the constraint ## m(t) = 2000 + 10 t ##, we get
## \frac{dv_f}{dt} = \frac{ 50 } {2000 + 10 t} ##

So,
## v_f(t) = v_f(t) - v_f(0) = 5 ( \ln(2000 + 10t) - \ln(2000) ) = 5 \ln(\frac{2000+ 10t}{2000 }) ##

And after 100 seconds, ## v_f(100) = 5\ln(1.5) \approx 2 ## meters per second.

Is that correct ?
 
Physics news on Phys.org
  • #2
Your work looks very good to me. I think your answer is correct.
 
  • #3
:-) Thank you very much
 

Similar threads

Replies
30
Views
5K
Replies
2
Views
1K
Replies
12
Views
6K
Replies
6
Views
3K
Replies
13
Views
3K
Back
Top