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**1. Homework Statement**

A 4.5 kg metal sphere is released in a fluid where k = 10.5 N s^2/m^s. How long does it take to reach 90% of its terminal velocity?

**2. Homework Equations**

Force of drag = kv^2, where k is the drag coefficient

(I believe we're not considering buoyancy.)

**3. The Attempt at a Solution**

ƩF = ma

a = (mg-kv^2)/m

a = [(4.5)((9.8) - (10.5)v^2]/4.5

a = 9.8 - 2.3v^2

At terminal velocity,

ƩF = 0

mg = kv^2

v = √(mg/k)

v = √[(4.5)(9.8)/10.5]

v = 2.0 m/s [down]

90% of this is 1.8 m/s [down].

So I know I have to find the time taken for the ball to achieve a velocity of 1.8 m/s^2 [down], and a have an equation with both acceleration and velocity. However, acceleration is not constant, so all of my kinematics knowledge (the constant acceleration equations) are useless, so I don't know how to proceed. We're supposed to create a graph to get the answer, but my teacher said there's a way to do this without graphing. I'm trying to find out what this method is. Thanks!

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