How do I find the terminal speed of a body in water?

[Ushitha Dissanayake]

Homework Statement

A baseball has a terminal speed of 42 m/s in air (ρ = 1.2 kg/m^3). What would be its terminal speed in water (ρ = 1.0 x 10^3 kg/m^3)?
A) 0.05 m/s
B) 1.5 m/s
C) 18 m/s
D) 42 m/s
E) 1200 m/s

Homework Equations

D=½Cρ*Av^2
where the magnitude of the drag force (D), relative speed = v, drag co-efficient = C, ρ is the air density (mass per volume) and A is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the
velocity )

v_t=√2*F_g/Cρ*A
where: v_t is the terminal speed, F_g is force gravity, and C, ρ and A is the same as above

The Attempt at a Solution

Im not sure how to solve this at all, the example that my book gave me involved finding terminal speed by giving me the drag co-efficient as well as the R to find the volume and area for a sphere

 

[Orodruin]

I believe this is a badly constructed problem because the obvious way to "solve" it (i.e., comparing the force of gravity on the ball with the drag force of water) is wrong and the proper way of solving it (also taking the buoyancy of the ball into account) requires more information than what is given in the problem (i.e., the density of the ball itself).

Either way, let us solve it the wrong way, which I suspect is what the problem constructor intended. Work symbolically, i.e., do not look for numerical values of middle steps until you reach a final expression. What would be the relation between the terminal velocities in air and water, respectively?

 

[Ushitha Dissanayake]

I'm not following

 

[Orodruin]

I'm not following

This is not helpful. Exactly what are you not following?

 

[Ushitha Dissanayake]

I don't really understand what you said

 

[Orodruin]

Still not helpful. You need to be specific in what it was you did not understand and why.

 

[Ushitha Dissanayake]

"What would be the relation between the terminal velocities in air and water, respectively?" I don't understand what you're implying here

 

[Orodruin]

What is the terminal velocities in water and air, respectively? What is the ratio between them? You have already given an expression for the terminal velocity so this should not be hard.

 

[Ushitha Dissanayake]

i tried finding a ratio by dividing (ρ = 1.2 kg/m^3) by (ρ = 1.0 x 10^3 kg/m^3) which gave 1.2*10^-3, then multiplying that with the terminal velocity of it in the air which gave 0.0504/ but the answer is 1.5

 

[Orodruin]

You are not listening. You cannot just go around randomly constructing ratios and expect other ratios to behave the same way. I asked you to find a very specific ratio by dividing the two expressions you have for the terminal velocity in air and water, respectively.

 

[Ushitha Dissanayake]

can you explain why we have to do it this way i still don't understand how this ratio thing works

 

[Orodruin]

A ratio is one quantity divided by another. Please use the analytical expressions you have, do not insert numbers. What do you get? It will be much clearer to you if you do it.

 

[Ushitha Dissanayake]

thanks for ur explanation

 

[Orodruin]

This still does not show that you have gotten the problem correctly. Please write out your solution. Also, as already mentioned, the ”correct” solution relies on false assumptions and we have not yet begun to discuss that. You will not learn properly if you are happy just getting the ”right” answer in these cases. You need to examine the problem assumptions.