Respuesta :
Answer:
[tex]t=\tau(\tanh^{-1}(\frac{v}{v_t}))[/tex]
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 [tex]v[/tex] of a falling object expressed in terms of terminal velocity [tex]v_t[/tex] and time 't' is given as:
[tex]v=v_t\tanh(\frac{t}{\tau})[/tex]
Where, [tex]\tau[/tex] is a constant and has units of time.
Now, expressing the above in terms of time 't', we get:
[tex]t=\tau(\tanh^{-1}(\frac{v}{v_t}))[/tex]
Therefore, the formula to find time to approach terminal velocity is:
[tex]t=\tau(\tanh^{-1}(\frac{v}{v_t}))[/tex]
When the velocity is 76% of terminal velocity, the time taken is [tex]\tau[/tex].
When the velocity is 96% of terminal velocity, the time taken is [tex]2\tau[/tex].
When the velocity is 99.5% of terminal velocity, the time taken is [tex]3\tau[/tex] and so on..
