# Terminal Velocity of a Ball Given Velocity and Acceleration

1. Dec 17, 2012

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.

2. Dec 17, 2012

### SammyS

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

3. Dec 17, 2012

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.

4. Dec 17, 2012

### SammyS

Staff Emeritus
bv = mg - ma .

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

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

5. Dec 17, 2012

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.)

6. Dec 17, 2012

### 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.

Last edited: Dec 17, 2012
7. Dec 17, 2012

### SammyS

Staff Emeritus
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.

8. Dec 17, 2012

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

9. Dec 17, 2012

### 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.

10. Dec 17, 2012

### SammyS

Staff Emeritus
That works too.

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