Terminal Velocity of a Ball Given Velocity and Acceleration

blradlof

1. The problem statement, all variables and given/known data
A steel ball is dropped from a great height. When its velocity is 20m/s its acceleration is 7.35 m/s^2. What is it's terminal velocity?


2. Relevant equations
V - V_o = at
F = ma
(Air resistance isn't given, so I don't think drag force can be used.)


3. The attempt at a solution
1. I know the final net force must equal 0.
2. The answer is 40 m/s, but I am unsure of how to get this answer.
3. I tried calculating drag force and taking the limit as time approaches infinity, but I must be making an error.

SammyS

How about an assumption such as air resistance is proportional to velocity or perhaps it's proportional to v² .

blradlof

Well I know air resistance is proportional to velocity, but I don't see how I can find air resistance without knowing surface area, mass, density, temperature, etc.

SammyS

bv = mg - ma .

You can find b/m for the given conditions. Then, what does v have to be if a = 0 ?


BTW: Are you sure about the answer of 40 m/s ?

If so, then I conclude that the air resistance follows a different law .

blradlof

v = mg/b

1. What does b stand for?
2. I'm sure it is 40 m/s. I came up with multiple answers, but they were all smaller than 40 m/s.
3. They don't give a value for mass. (If I understand what b stands for, I may be able to figure it out.)

ehild

The forces acting on the falling ball are gravity and air resistance, which is proportional to velocity and opposite to it.
Writing up Newton's second law, F=ma=mg-kv. The acceleration is
a=g-(k/m) v.
You know that a=7.35 m/s² when v=20 m/s. Plug in: you will find k/m.

The ball will reach the terminal velocity when the two forces - gravity and air resistance - cancel, that is, the acceleration is zero: g-(k/m) v=0. You know k/m, how big is v?
If the given result is 40 m/s then the air resistance is taken proportional to v².
Do the same procedure, replacing v by v².

SammyS

b is simply a constant of proportionality.

If you assume that the drag force (air resistance) is proportional to v², you will get 40 m/s for your answer.

This whole thing is essentially just working with proportiona.

blradlof

Thank you so much! I forgot that air resistance can be proportional to v and v^2.

ReachingFoul

a=g-kv, a is a function of veolcity(assumption) k is a constant and then we can find k, by using given data.

then find a=0, which kv=g, you can find out v which is terminal veolcity.

SammyS

That works too.

In the given problem it appears that acceleration, as well as force, is proportional to v² .