Fluid dynamics+calculus: picking which vars to integrate

[white 2.5rs]

Homework Statement

This is extra credit, and the test has passed (no one got it), the prof said he'll still take efforts on it so here is my attempt.

A rocket sled is going down a test track at 180 km/hr (calculated to 50 m/s). It drops a snorkel into a trough of water. This diverts 30 kg/s of water vertically. The sled has mass of 500kg (including the snorkel and water in it).

The question is how far will the sled travel once it drops the snorkel.

I know I need to integrate from 50 to 0... but beyond that I don't know what vars to integrate in order to get something to plug into kinematic equations...

negate fluid+track+air friction.

Homework Equations

The Attempt at a Solution

I know to integrate from 50 to zero... but i don't know which vars to integrate

 

[gabbagabbahey]

Ask yourself what is causing the sled to slow down (lose momentum)...What rate does that 'something' decrease the sled momentum at?

Also, are you told the angle that the sled makes with the vertical or anything about the shape of the snorkel?

 

[white 2.5rs]

The sled slows due to the resultant force associated with shooting water 90 deg from the horizontal.

The shape/angle aren't important as far as the class knows. Only knowing flow @ 180km/hr is important.

 

[gabbagabbahey]

The sled slows due to the resultant force associated with shooting water 90 deg from the horizontal.

The shape/angle aren't important as far as the class knows. Only knowing flow @ 180km/hr is important.

The shape/angle are actually very important! However, since you are not told anything about them you will need to make some simplifying assumptions. I would assume that the track is horizontal if not told otherwise. And I would assume that the shape of the snorkel is such that all of the force of the water that is ejected is directed opposite to the sled's motion. (In general, only a fraction of that force will actually be directed opposite the sleds motion)

Anyways, what is the rate of change of the momentum of the water being forced through the snorkel at time t (in the rest frame of the track/water) if the sled's speed at time t is v(t) (Remember, momentum is a vector!)? What acceleration a(t) does that cause the sled?

 

[white 2.5rs]

this is only a quick response but the angle is at 90 deg
the shape is to be neglected
the change in momentum does change as it is integrated from 50 to 0 but i am still unfamiliar w/ which vars to integrate

 

[gabbagabbahey]

the change in momentum does change as it is integrated from 50 to 0 but i am still unfamiliar w/ which vars to integrate

Start by finding a relationship between the speed of the boat at a given instant and the rate of change of its momentum at that instant:

Anyways, what is the rate of change of the momentum of the water being forced through the snorkel at time t (in the rest frame of the track/water) if the sled's speed at time t is v(t) (Remember, momentum is a vector!)? What acceleration a(t) does that cause the sled?