Terminal Velocity of a Ball Given Velocity and Acceleration

In summary: This can be figured out by replacing v with v2 in the equation for F. This gives us the answer of 40 m/s.
  • #1
blradlof
4
0

Homework Statement


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?


Homework Equations


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


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.
 
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  • #2
blradlof said:

Homework Statement


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?


Homework Equations


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


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.
How about an assumption such as air resistance is proportional to velocity or perhaps it's proportional to v2 .
 
  • #3
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
blradlof said:
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.
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 .​
 
  • #5
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
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:
  • #7
blradlof said:
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.)
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
Thank you so much! I forgot that air resistance can be proportional to v and v^2.
 
  • #9
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
ReachingFoul said:
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.
That works too.

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

1. What is terminal velocity?

Terminal velocity is the maximum velocity that a falling object can reach when the drag force of the surrounding air is equal to the force of gravity.

2. How is terminal velocity calculated?

The formula for terminal velocity is Vt = √(2mg/ρACd), where Vt is the terminal velocity, m is the mass of the object, g is the acceleration due to gravity, ρ is the density of the air, A is the cross-sectional area of the object, and Cd is the drag coefficient.

3. Does the shape of the object affect its terminal velocity?

Yes, the shape of the object does affect its terminal velocity. Objects with a larger surface area, such as a parachute, will reach a lower terminal velocity compared to objects with a smaller surface area, such as a bullet.

4. How does air resistance affect terminal velocity?

Air resistance, also known as drag force, is what causes an object to reach its terminal velocity. As an object falls, air resistance increases until it is equal to the force of gravity, resulting in the object maintaining a constant velocity.

5. Can the terminal velocity of an object change?

Yes, the terminal velocity of an object can change if there is a change in the object's mass, surface area, or the density of the air it is falling through. For example, a lighter object will have a lower terminal velocity than a heavier object with the same surface area and falling through the same air density.

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