suppose an object of weight 128 lbs is projected downwards with an initial velocity of 10ft/s in medium that offers resistance of magnitude 8|v|. Assuming gravitational acceleration is constant (g=32ft/s^2) find the velocity at time t.
suppose an object of weight 128 lbs is projected downwards with an initial velocity of 10ft/s in medium that offers resistance of magnitude 8|v|. Assuming gravitational acceleration is constant (g=32ft/s^2) find the velocity at time t.
Defining positive to be upward:
F = ma = 8|v| - mg
v is always directed downward. (A resistance can't propel the object upward, after all.) So...
m*dv/dt = -8v - mg
dv/dt + (8/m)v = g with v(0) = -10.
The homogeneous equation is
dv_h/dt + (8/m)v_h = 0
So
v_h(t) = Ae^{-(8/m)t}
And the particular solution looks like it's
v_p(t) = B
Putting this into the differential equation:
dv_p/dt + (8/m)v_p = g
0 + (8/m)B = g
B = mg/8
Thus
v(t) = v_h(t) + v_p(t) = Ae^{-(8/m)t} + (mg/8)
Now, v(0) = -10, so
-10 = A + (mg/8)
A = -10 - (mg/8) = -(mg + 80)/8
Thus
v(t) = -(1/8)(mg + 80)e^{-(8/m)t} + (mg/8)
-Dan