Question

What is the formula for finding the time it takes for an object to reach terminal velocity?

Answer 1

Answer:

$t=\tau(\tanh^{-1}(\frac{v}{v_t}))$

Explanation:

Terminal velocity is the maximum velocity attained by a body when the net upward force on it is equal to the net downward force as it falls through a fluid from a certain height.

The time required to reach terminal velocity is infinite as the object never reaches absolute terminal velocity and always approaches terminal velocity.

The velocity $v$ of a falling object expressed in terms of terminal velocity $v_t$ and time 't' is given as:

$v=v_t\tanh(\frac{t}{\tau})$

Where, $\tau$ is a constant and has units of time.

Now, expressing the above in terms of time 't', we get:

$t=\tau(\tanh^{-1}(\frac{v}{v_t}))$

Therefore, the formula to find time to approach terminal velocity is:

$t=\tau(\tanh^{-1}(\frac{v}{v_t}))$

When the velocity is 76% of terminal velocity, the time taken is $\tau$.

When the velocity is 96% of terminal velocity, the time taken is $2\tau$.

When the velocity is 99.5% of terminal velocity, the time taken is $3\tau$ and so on..