Setting up an equation for max efficiency

[GordonF]

Can someone please help me get started or tell me how to tackle this problem. (or tell me if I posted this in the wrong section). I need to set up equation that relates three variables and allows me to find out what the minimum value can be for one, given the other two.

This is the scenario:

I have a liquid that I inject through a tube at a certain rate (cm/sec). When I stop the input of liquid, it will continue to move through the tube at that rate with a distance from the beginning of the liquid to the end that is equal to the time I spent injecting it. This is Variable A.

Then I have a camera that is intended to trace this liquid as it moves down the tube. The camera will move at a different rate down the tube but it always has to be within the range of the beginning of the liquid to the end of the liquid. The speed at which the camera moves is Variable B. (cm/sec)

Variable C is the time interval that I wish to observe the liquid with my camera. (sec)

If the camera speed and desired observation time is given to me, what is the minimum amount of time I need to spend injecting my liquid so that at any point during the given observation time, my camera is observing some of that liquid? the camera can start moving at any point after or on t=0.

 

[JeffM]

Can someone please help me get started or tell me how to tackle this problem. (or tell me if I posted this in the wrong section). I need to set up equation that relates three variables and allows me to find out what the minimum value can be for one, given the other two.

This is the scenario:

I have a liquid that I inject through a tube at a certain rate (cm/sec). When I stop the input of liquid, it will continue to move through the tube at that rate with a distance from the beginning of the liquid to the end that is equal to the time I spent injecting it. This is Variable A.

Then I have a camera that is intended to trace this liquid as it moves down the tube. The camera will move at a different rate down the tube but it always has to be within the range of the beginning of the liquid to the end of the liquid. The speed at which the camera moves is Variable B. (cm/sec)

Variable C is the time interval that I wish to observe the liquid with my camera. (sec)

If the camera speed and desired observation time is given to me, what is the minimum amount of time I need to spend injecting my liquid so that at any point during the given observation time, my camera is observing some of that liquid? the camera can start moving at any point after or on t=0.

I'd help if I could, but right now I cannot because I have no clue what you are saying. How can a distance equal a time? You talk about cm per sec. That is obscure to me. What is a centimeter of a liquid? I could understand cubic centimeters per second, but then I suspect I would need to know the tube's cross-sectional area in square centimeters to compute a rate of lineal propagation in centimeters per second. So I do not get even the math of the problem. There are physicists here who may have additional questions. For example, I have a suspicion that, unless the liquid has such high viscosity as to be almost solid, gravitational pressure will make the liquid propogate along the tube at a rate higher than warranted by the velocity of injection while, after the injection ceases, friction will lower the rate of propagation. (I recollect that gravity and friction are forces frequently considered relevant in physics, but I may be wrong. I was taught physics by the football coach, who had to be kept occupied until practice time, and I learned virtually nothing.)

 

[DrPhil]

Can someone please help me get started or tell me how to tackle this problem. (or tell me if I posted this in the wrong section). I need to set up equation that relates three variables and allows me to find out what the minimum value can be for one, given the other two.

This is the scenario:

I have a liquid that I inject through a tube at a certain rate (cm/sec). When I stop the input of liquid, it will continue to move through the tube at that rate with a distance from the beginning of the liquid to the end that is equal to the time I spent injecting it. This is Variable A.

Then I have a camera that is intended to trace this liquid as it moves down the tube. The camera will move at a different rate down the tube but it always has to be within the range of the beginning of the liquid to the end of the liquid. The speed at which the camera moves is Variable B. (cm/sec)

Variable C is the time interval that I wish to observe the liquid with my camera. (sec)

If the camera speed and desired observation time is given to me, what is the minimum amount of time I need to spend injecting my liquid so that at any point during the given observation time, my camera is observing some of that liquid? the camera can start moving at any point after or on t=0.

If we make sense out of the units, it may help us to see what is being asked.

The tube has a uniform cross section, so volume per second can be reduced to just the linear velocity. Both the left edge and the right edge are moving "at a certain rate," which has units of cm/s. Call that speed V1. If A is the duration of injection, then the length of the injected fluid is A*V1, which you can see has units of (s)(cm/s) = (cm). The time it takes the length of liquid to pass a stationary point is A.

Does the problem say that you want to see every part of the fluid within the time C? If C is less than A, you will want the camera motion to be opposite the flow direction to compress the observation time. If C is longer than A, then moving the camera in the same direction as the flow will expand the time.You can make a quantity with units of velocity by dividing length A*V1 by time C: V2 = v1*(A/C). How does that relate to velocity B?

We can't tell where you are stuck unless you show us your work.

 

[GordonF]

If we make sense out of the units, it may help us to see what is being asked.

The tube has a uniform cross section, so volume per second can be reduced to just the linear velocity. Both the left edge and the right edge are moving "at a certain rate," which has units of cm/s. Call that speed V1. If A is the duration of injection, then the length of the injected fluid is A*V1, which you can see has units of (s)(cm/s) = (cm). The time it takes the length of liquid to pass a stationary point is A.

Does the problem say that you want to see every part of the fluid within the time C? If C is less than A, you will want the camera motion to be opposite the flow direction to compress the observation time. If C is longer than A, then moving the camera in the same direction as the flow will expand the time.You can make a quantity with units of velocity by dividing length A*V1 by time C: V2 = v1*(A/C). How does that relate to velocity B?

We can't tell where you are stuck unless you show us your work.

Thanks for the feedback. I understand the question is very vague. It has been given to me as a practical problem that I tried to rephrase (not so successfully) into math terms.

This is correct - "The tube has a uniform cross section, so volume per second can be reduced to just the linear velocity. Both the left edge and the right edge are moving "at a certain rate," which has units of cm/s. Call that speed V1. If A is the duration of injection, then the length of the injected fluid is A*V1, which you can see has units of (s)(cm/s) = (cm). The time it takes the length of liquid to pass a stationary point is A."

The point is not to observe every part of the fluid during the time interval C, but to make sure that as the camera has moved for C seconds it has always had a part of the length of the injected fluid (A*V1) in its line of sight.

C(time interval) * V2(velocity of camera) will always be greater than the length of the injected fluid (A*V1)
V1(velocity of the fluid) will always be greater than the V2(velocity of camera).

The problem is how long do I have to make length of the injected fluid (A*V1) so that as it passes through the part of the tube that I want to observe (C*V2), my camera always has some part of that liquid in its view. I want to know how short I can make the duration of injection (A), if V1, V2, and C are known.

I really appreciate the help, please tell me if I have to clarify better.