Distance to reach terminal velocity

In summary, we have a body with a mass of 100 Kg falling towards the ground and the goal is to determine the distance and time it takes to reach terminal velocity. The relevant equations are a=dv/dt, v=ds/dt, drag force=100v^2, and F=ma. The solution involves using differential equations to solve for time and distance. However, when using numerical values for the variables, such as mass and drag coefficient, it is important to keep them as symbols rather than inserting numerical values too early. The easiest way to solve for distance is by using the equation dv/dt = (dv/ds)(ds/dt) = v dv/ds, which gives a simple integral. However, for
  • #1
abdo799
169
4

Homework Statement



A body with a 100 Kg of mass is falling , what's the distance and time to reach terminal velocity?

Homework Equations


a=dv/dt v=ds/dt drag force= 100v^2
F=ma

The Attempt at a Solution


I don't really need the numbers , just the expressions
I don't know how to write the equations so I am going to capture my attempt and post as photos. I don't know if I am right or wrong yet. The photos were mixed during the upload , the last photo is the first one and vice versa
 

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  • #2
Sorry, x = infinity, t = infinity.

The terminal velocity is a finite number but the distance and time to reach it are asymptotes which are never completely reached.
 
  • #3
rude man said:
Sorry, x = infinity, t = infinity.

The terminal velocity is a finite number but the distance and time to reach it are asymptotes which are never completely reached.

yea i know , but we can get an approximation , i could use excel to do it , but i saw this thread ( https://www.physicsforums.com/showthread.php?t=502933 ) he used differential equations to solve for time , so i thought maybe i could use it to solve for distance
 
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  • #4
abdo799 said:
yea i know , but we can get an approximation , i could use excel to do it , but i saw this thread ( https://www.physicsforums.com/showthread.php?t=502933 ) he used differential equations to solve for time , so i thought maybe i could use it to solve for distance

You tell me what constitutes "reaching terminal velocity", like for example 99% of terminal velocity, and maybe I can show you how to get that.
 
  • #5
rude man said:
You tell me what constitutes "reaching terminal velocity", like for example 99% of terminal velocity, and maybe I can show you how to get that.
Thanks , i did a spreadsheet and i got the answers , my question is : is this calculus approach wrong?
 
  • #6
abdo799 said:
Thanks , i did a spreadsheet and i got the answers , my question is : is this calculus approach wrong?
For the time, yes, though my preference would be to keep all the variables as symbols (m for mass, D for drag coefficient etc.) rather then inserting numerical values som early.
You can solve the time integral using partial fractions.
For the distance, the easiest way is to use dv/dt = (dv/ds)(ds/dt) = v dv/ds. That gives you a very easy integral.
 
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  • #7
haruspex said:
For the time, yes, though my preference would be to keep all the variables as symbols (m for mass, D for drag coefficient etc.) rather then inserting numerical values som early.
You can solve the time integral using partial fractions.
For the distance, the easiest way is to use dv/dt = (dv/ds)(ds/dt) = v dv/ds. That gives you a very easy integral.

So integration does work ? thanks for that ultra-easy integral :thumbs: , the one i figured out was a nightmare
 
  • #8
rude man said:
You tell me what constitutes "reaching terminal velocity", like for example 99% of terminal velocity, and maybe I can show you how to get that.

if i had a 50 m/s terminal velocity for example , if i put v= 49 m/s , i can work it out with calculus right?
 
  • #9
abdo799 said:
if i had a 50 m/s terminal velocity for example , if i put v= 49 m/s , i can work it out with calculus right?

Right!
 
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  • #10
Thanks
 
  • #11
One last question before we call it a day , when i use differential equation calculator in wolfram , he used log instead of ln in the integration , he gave me t in terms of v . when he was trying to put v the subject ( to put v in terms of t ) he used e . How come?
 
  • #12
abdo799 said:
One last question before we call it a day , when i use differential equation calculator in wolfram , he used log instead of ln in the integration , he gave me t in terms of v . when he was trying to put v the subject ( to put v in terms of t ) he used e . How come?

What was your diff. equation?
 
  • #13
rude man said:
What was your diff. equation?

Int(t)=int(dv/(228-0.113v^2))
 
  • #14
abdo799 said:
Int(t)=int(dv/(228-0.113v^2))

OK, you mean dt = dv/(228 - 0.113 v2).

That "228" bothers me. I have g for that constant = 9.81 ms-2.
Also, you need to know the drag coefficient = drag force/v2. I saw nothing in your problem that gave that coefficient a number. Where did you get the number 0.113 from, which has to be the drag coefficient divided by the mass of the diver?
 
  • #15
rude man said:
OK, you mean dt = dv/(228 - 0.113 v2).

That "228" bothers me. I have g for that constant = 9.81 ms-2.
Also, you need to know the drag coefficient = drag force/v2. I saw nothing in your problem that gave that coefficient a number. Where did you get the number 0.113 from, which has to be the drag coefficient divided by the mass of the diver?

This is not really about skydiving, i am pretty sure the other calculation are correct, the body is teardrop shape which according to google has a coefficient of drag of 0.05
 
  • #16
OK, but I still think you should show how you got those two numbers: 228 and 0.113.

If you're sure they're right (and I don't think they are), then go ahead and show what your solution was for v(t) or t(v) or both.
 

FAQ: Distance to reach terminal velocity

What is terminal velocity?

Terminal velocity is the maximum speed that an object can reach when falling through a fluid, such as air or water. At terminal velocity, the object's weight is balanced by the drag force of the fluid, resulting in a constant velocity.

How is terminal velocity calculated?

The formula for terminal velocity is Vt = sqrt((2mg)/pAC), where Vt is the terminal velocity, m is the mass of the object, g is the acceleration due to gravity, p is the density of the fluid, A is the cross-sectional area of the object, and C is the drag coefficient.

Does distance affect terminal velocity?

Yes, the distance an object falls can affect its terminal velocity. As an object falls, it gains speed and air resistance increases. Once the drag force equals the object's weight, the object reaches terminal velocity. The longer the distance, the more time an object has to accelerate and reach terminal velocity.

How does air density affect terminal velocity?

Air density does affect terminal velocity. The higher the air density, the greater the drag force on the falling object, leading to a lower terminal velocity. Conversely, in lower air density, there is less drag force and the object can reach a higher terminal velocity.

Can an object reach terminal velocity in a vacuum?

No, an object cannot reach terminal velocity in a vacuum because there is no fluid for the object to fall through and experience drag force. Without air resistance, the object will continue to accelerate until it reaches a constant speed or collides with another object.

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