Does an Object Ever Truly Reach Terminal Velocity?

[Ali Asadullah]

We have the following formula for the velocity of an object moving in a fluid.
v = v_t -v_te^-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=kv_t which shows that object achieves terminal velocity?
Can anyone please explain this seemingly contradiction?

 

[berkeman]

We have the following formula for the velocity of an object moving in a fluid.
v = v_t -v_te^-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=kv_t 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...

 

[Ali Asadullah]

But sir v_t=v only when time t = infinity.
and hence v can never equal to v_t.
But while deriving it we have supposed that we have achieved v_t.

 

[Stonebridge]

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.

 

[dacruick]

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.

 

[Ali Asadullah]

Thanks a lot Sir Stonebridge and Sir dacruick.

 

[GRDixon]

We have the following formula for the velocity of an object moving in a fluid.
v = v_t -v_te^-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=kv_t 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.