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## Homework Statement

For small, slowly falling objects, the assumption made in equation (1) (see below) is good. For larger, more rapidly falling objects, it is more accurate to assume that the magnitude of the drag force is proportional to the square of the velocity with the force orientation opposite to that of velocity.

Write a differential equation for the velocity of a falling object of mass [itex]m[/itex] if the magnitude of the drag force is proportional to the square of the velocity

## Homework Equations

(1) [itex]m \frac{dv}{dt} = -mg - \gamma v[/itex]

[itex] \gamma > 0[/itex]

## The Attempt at a Solution

Since the drag force ([itex]- \gamma v[/itex]) is proportional to the square of the velocity:

[itex] | - \gamma v | = \gamma | v | \propto v^2[/itex]

and since the orientation is opposite to that of velocity

[itex] \gamma | v | = -kv^2 [/itex] where [itex]k[/itex] is some constant.

but this is where I get stuck.

The answer in the back of the book has the drag force as [itex]- \gamma v |v|[/itex].

Where did I go wrong?

Thank you anyone for your help!