Kinematic and geometric similarity (fluids)

[selig5753]

Homework Statement

In filming a motion picture, an automobile of length L is to be driven over a vertical cliff at speed V. The normal filming speed is N frames per second. If a model car of length l is substituted for economy what is the proper value for the model speed v and filming speed n? Assume that the film will be projected at normal speed and that air resistance can be neglected; i.e., the mechanics are essentially those of a point mass

Relevant Equations

v = V sqrt(l/L), n = N sqrt(L/l)

My attempt at a solution is to start off first denoting V_a to be the automobile an V_e to be the economy version. Same goes with l_a and l_e. To try and relate the two I have tried: V_a I_a = V_l L_e, however I am really not sure how they got the square root.

The answer is: v = V sqrt(l/L), n = N sqrt(L/l)

 

[Master1022]

Homework Statement:: In filming a motion picture, an automobile of length L is to be driven over a vertical cliff at speed V. The normal filming speed is N frames per second. If a model car of length l is substituted for economy what is the proper value for the model speed v and filming speed n? Assume that the film will be projected at normal speed and that air resistance can be neglected; i.e., the mechanics are essentially those of a point mass
Relevant Equations:: v = V sqrt(l/L), n = N sqrt(L/l)

My attempt at a solution is to start off first denoting V_a to be the automobile an V_e to be the economy version. Same goes with l_a and l_e. To try and relate the two I have tried: V_a I_a = V_l L_e, however I am really not sure how they got the square root.

The answer is: v = V sqrt(l/L), n = N sqrt(L/l)

Just so we are clear:
- Geometric Similarity - the model must be the same shape as the prototype but can be scaled to be larger or smaller
- Kinematic Similarity - the velocity at any point in the fluid flow must be proportional (by a constant scale factor) to the velocity at the corresponding point in the prototype field

Okay, so for these sorts of problems, we usually want to relate dimensionless groups (e.g. Reynold's number) to one another. Have you been given any dimensionless groups? If not, have you learned about the 'Buckingham $\pi$ Theorem'? I would perhaps make a start by creating these dimensionless groups (or getting some from a list of standard ones) and seeing whether you can use the definitions above to set them equal to each other for the real car and the model car to get the required results.

I would encourage you to look into Buckingham $\pi$ if you haven't heard about it - it is a bit too much for me to type up on here, but basically the concept is taking all the variables you have in your problem and using them to create dimensionless groups by looking at their units.

 

[selig5753]

I have learned about the Buckingham Pi theorem however, I was not entirely sure if this was the correct approach since we are given velocity LT^{-1}, length L and that's it. Am I missing something?

My approach with the buckingham Pi theorem was this: $(LT^{-1})^{a} (L)^{b} (n T^{-1})^{c}$.