Sand-spraying locomotive

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In summary, a sand-spraying locomotive sprays sand into a freight car ahead of it at a constant distance and speed. The sand is transferred at a rate of 10 kg/s with a velocity of 5 m/s relative to the locomotive. The initial mass of the car is 2000 kg and its final speed after 100 seconds is approximately 2 m/s. This is found using the conservation of momentum and the changing mass of the car.
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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 ?
 
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  • #2
Your work looks very good to me. I think your answer is correct.
 
  • #3
:-) Thank you very much
 

1. What is a sand-spraying locomotive?

A sand-spraying locomotive is a special type of train used to spread sand on the tracks to improve traction and prevent slippage.

2. How does a sand-spraying locomotive work?

A sand-spraying locomotive has a sand storage tank and a mechanism that disperses the sand onto the tracks as it moves. The sand creates a rough surface, helping the train's wheels grip the tracks better.

3. Why is sand necessary for locomotives?

Sand is necessary for locomotives because it helps improve traction on the tracks. Trains can experience slippage on the tracks, especially in wet or icy conditions, and spreading sand helps prevent accidents and delays.

4. Are there any environmental concerns with sand-spraying locomotives?

While sand-spraying locomotives do contribute to air pollution by releasing small particles into the air, they are necessary for safe train operation. However, some railways are now using alternative technologies, such as electric or magnetic adhesion systems, to reduce the use of sand.

5. What are the benefits of using a sand-spraying locomotive?

The main benefit of using a sand-spraying locomotive is improved safety and efficiency. By spreading sand on the tracks, the train is less likely to experience slippage and delays, which can be costly. It also helps reduce wear and tear on the train's wheels and tracks.

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