Terminal Speed, AP Physics

Question

a 9kg object falls from rest through a viscous medium and experiences a resistive force R=-bv. if the object's speed reaches 1/2 of its terminal speed at 5.54 seconds, find the terminal speed.

Answer 1

Since the resistive force is linear (R = -bv), you can derive the following equation using differential calculus:

 

v = (mg/b)[1 - exp(-bt/m)]

 

(I assume you were given this equation, but I can derive it in detail if you need...)

 

Next you need to solve for the constant "b":

 

As t goes to infinity, v_t = mg/b

 

The above equation for b is now:

 

0.5v_t = v_t[1 - exp(-b(5.54s)/9kg)]

 

Solve this for b to obtain b = 1.13 kg/s

 

Now that you know b, use the equation for when t goes to infinity:

 

v_t = mg/b

 

v_t = (9kg)(9.8 m/s²)/(1.13 kg/s)

 

v_t = 78.3 m/s