Does an Object Ever Truly Reach Terminal Velocity?

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Discussion Overview

The discussion centers around the concept of terminal velocity and whether an object can truly reach this state when moving through a fluid. Participants explore the mathematical implications of the relevant formulas and the physical interpretations of these equations, examining both theoretical and practical aspects.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a formula for velocity, suggesting it indicates that an object never achieves terminal velocity, while another formula implies it does.
  • Another participant questions the meaning of "never," arguing that in exponential equations, values can get arbitrarily close to terminal velocity without actually reaching it.
  • A different participant asserts that terminal velocity is only reached as time approaches infinity, thus implying that in practical terms, it can never equal terminal velocity.
  • One participant draws parallels to other physical laws, such as radioactive decay and Newton's law of cooling, to illustrate that certain values approach limits without ever reaching them.
  • Another participant provides an example involving temperature change, emphasizing that two objects will never reach the same temperature except at infinity, highlighting the concept of asymptotic behavior.
  • A later reply clarifies that while the formulas indicate that an object approaches terminal velocity, it does so asymptotically and never actually reaches it when starting from rest.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of terminal velocity, with some arguing that it can never be reached while others suggest it can be approached closely enough in practical scenarios. The discussion remains unresolved regarding the implications of these mathematical models in real-world contexts.

Contextual Notes

The discussion highlights the limitations of mathematical models in capturing real-world phenomena, particularly the assumptions involved in defining terminal velocity and the conditions under which these models apply.

Ali Asadullah
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We have the following formula for the velocity of an object moving in a fluid.
v = vt -vte-kt/m.
This formula shows that an object can never achieve terminal velocity but in the derivation of this formula given in "Fundamental of Physics" by H D Young and Freedman 10 edition we use the formula mg=kvt which shows that object achieves terminal velocity?
Can anyone please explain this seemingly contradiction?
 
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Ali Asadullah said:
We have the following formula for the velocity of an object moving in a fluid.
v = vt -vte-kt/m.
This formula shows that an object can never achieve terminal velocity but in the derivation of this formula given in "Fundamental of Physics" by H D Young and Freedman 10 edition we use the formula mg=kvt which shows that object achieves terminal velocity?
Can anyone please explain this seemingly contradiction?

What do you mean by "never"? In an exponential equation like that, you only have to go out a few time constants to get arbitrarily close to the final value...
 
But sir vt=v only when time t = infinity.
and hence v can never equal to vt.
But while deriving it we have supposed that we have achieved vt.
 
There are a number of laws in physics that have an exponential like this one. Radioactive decay is one. Another is the rate of loss of heat from a hot body. In both of these cases, the maths says that the value never reaches zero because t would need to be infinity. In that you are perfectly correct.
However, this is the maths. In the real world, we find that we don't need to wait until infinity before the value falls to such a low level that it is so near to zero that the difference cannot be measured, or the difference is of no significance.
 
Yes and just to add an example to what stronebridge is saying. Newton's law of heating and cooling which you may be familiar with or population modelling. These both have similar concepts behind them. If you put a steel ball in a 30F room. and The steel ball is at 0F, they will never become the same temperature except at infinity. This is because there is a point where the steel ball is 29.999999999999999999999999999999999999999999999999999999 F, and since the law of heating and cooling depends on difference in temperature, it will take forever for them to be equal.
 
Thanks a lot Sir Stonebridge and Sir dacruick.
 
Ali Asadullah said:
We have the following formula for the velocity of an object moving in a fluid.
v = vt -vte-kt/m.
This formula shows that an object can never achieve terminal velocity but in the derivation of this formula given in "Fundamental of Physics" by H D Young and Freedman 10 edition we use the formula mg=kvt which shows that object achieves terminal velocity?
Can anyone please explain this seemingly contradiction?

The first formula is stating that if the object is released from a state of rest, and allowed to accelerate downward through a viscous fluid in a gravitational field, then its velocity at time t will be as specified. The second formula is stating that, AT terminal velocity, the downward force of gravity WOULD be equal and oppositely directed to the drag force (so that they sum to zero). But when the initial velocity is less than the terminal velocity, then the object never achieves a velocity equal to terminal velocity. Its velocity only approaches the terminal velocity asymptotically.
 

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