Question

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

Answer 1

`t=\tau(\tanh^(-1)((v)/(v_t)))`

Step-by-step 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((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)((v)/(v_t)))`

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

`t=\tau(\tanh^(-1)((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..