# Velocity under a drag force

**1. The problem statement, all variables and given/known data**

An object moving in a liquid experiences a linear drag force: D⃗ =(bv, direction opposite the motion), where b is a constant called the drag coefficient. For a sphere of radius R, the drag constant can be computed as b=6πηR, where η is the viscosity of the liquid.

Find an algebraic expression for vx(t), the x-component of velocity as a function of time, for a spherical particle of radius R and mass m that is shot horizontally with initial speed v0 through a liquid of viscosity η.

**Express your answer in terms of the variables v0, η, R, t, m, and appropriate constants.**

**2. Relevant equations**

**3. The attempt at a solution**

Thinking of typical dynamics, I divided the drag force, bv by m to get the acceleration. Then I subtracted acceleration times time from the inivial velocity, v0. So it looked like this:

v0- (6πηRv0t)/m.

Obviously it wasn’t right, as my teacher today told me that I have to integrate the acceleration function to get the velocity function. I have no idea how to integrate the acceleration function in which it looks like every single variable are constants.

Help would be appreciated! Thanks in advance.

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