# Finding an expression for the velcoity v as a function of time.

#### alanjkim

Here's the problem:
Suppose an object with mass m is projected with initial velocity v 0 along the horizontal direction. Assume quadratic air resistance of the form F drag = αv^2 , where F drag is the magnitude of the drag force. Find an expression for the
velocity v as a function of time.
Here's what I have so far:
ma = αv^2
m(dv/dt) = αv^2
(dv/dt) =( αv^2/m)
v(t) = integral of (( αv^2/m))

Am I on the right track here? Can someone help me finish this problem.

#### topsquark

Forum Staff
Here's the problem:
Suppose an object with mass m is projected with initial velocity v 0 along the horizontal direction. Assume quadratic air resistance of the form F drag = αv^2 , where F drag is the magnitude of the drag force. Find an expression for the
velocity v as a function of time.
Here's what I have so far:
ma = αv^2
m(dv/dt) = αv^2
(dv/dt) =( αv^2/m)
You are good up to here. This is the Bernoulli differential equation. See here. (Set p(x) = 0.)

-Dan

#### alanjkim

You are good up to here. This is the Bernoulli differential equation. See here. (Set p(x) = 0.)

-Dan
So then getting the integral of (dv/dt) gets v(t) which is the left side of the equation. Any help on the right side? I'm very confused as to how to get the integral of the algebraic equation with multiple variables, if that is even how to go about the problem.

#### HallsofIvy

Here's the problem:
Suppose an object with mass m is projected with initial velocity v 0 along the horizontal direction. Assume quadratic air resistance of the form F drag = αv^2 , where F drag is the magnitude of the drag force. Find an expression for the
velocity v as a function of time.
Here's what I have so far:
ma = αv^2
m(dv/dt) = αv^2
(dv/dt) =( αv^2/m)
v(t) = integral of (( αv^2/m))

Am I on the right track here? Can someone help me finish this problem.
You are fine until the last line! You can write dv= (av^2/m)dt but v is an unknown function of t. You cannot integrate av^2/m with respect to t.

Instead write dv/v^2= a/m dt and integrate both sides, the left side with respect to v and the right side with respect to t.

• 1 person

#### topsquark

Forum Staff
You are fine until the last line! You can write dv= (av^2/m)dt but v is an unknown function of t. You cannot integrate av^2/m with respect to t.

Instead write dv/v^2= a/m dt and integrate both sides, the left side with respect to v and the right side with respect to t.
@alanjkim: Yes, do it this way. The equation is separable. I always do things the hard way!

-Dan