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## Homework Statement

A bullet is fired from a gun. As soon as it leaves the barrel, the bullet begins to decelerate due to air resistance at a rate defined by the following equation:

a = -kv

^{2}

The variable "a" is (negative) acceleration.

The variable "v" is instantaneous velocity.

The variable "k" is simply a constant that relates to the particular properties of the bullet.

**From this equation, write a function that describes the time it will take the bullet to travel x distance.**

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Now, I have asked this question before, and I actually was given a solution by someone at one point that works. It involved integrating the given function twice, but truthfully, I don't understand it at all, and I would really like to. I am hoping that someone on this forum might be able to help me understand the calculus involved with this sort of thing.

## The Attempt at a Solution

The solution is this:

t = (1/(V * k)) * (exp(D * k) - 1)

"V" is the initial velocity of the bullet (i.e., the muzzle velocity); "k" is the constant from the original equation; "D" is the distance the bullet traveled; and "exp" is just shorthand for the exponential equation (i.e., e to the power of D * k).

This solution works. I have tested it. But I have no idea how it was obtained, and it's driving me crazy. If anyone can help me understand it, I would be most appreciative!