Differential Equations Word Problem

[Jeff3355]

Given the equation: -m g + k v^2 = m a
Find:
A) The time it takes for a golf ball dropped from an elevation of 1000 feet to hit the ground.
B) Velocity of the golf ball when it hits the ground.
C) What is the terminal velocity of the golf ball?

m=mass of golf ball
g=32.17 ft/s^2
k=(Cd)(rho)(ball frontal area)/2 = 4.60 x10^-6
v=velocity
a=acceleration
W=0.10 lb (golf ball weight), where W=mg, or m=W/g

Assume drag coefficient (Cd) and air density (rho) are constant for simplicity, therefore k is constant.
The term k v^2 is the drag force due to wind acting upward when the ball falls downward (wind drag force is proportional to the square of the velocity).

Hints:
a = dv/dt
v = dy/dt, or the integral of v(t)=y(t) where y is the elevation.

[Guest]

Lets start with what I think is an easy bit:-----
The terminal velocity is when drag forces balance the gravity forces and the thing falls at constant speed, so has no net acceleration downwards, so....
mg = kv^2
0.1 = 4.6x10^-6 xv^2
v = sqrt(0.1 / 4.6x10^-6)
v = 147.44 feet per second is the terminal velocity

[Guest]

Without any more counting, if ball reaches terminal velocity on the way down which I assume it will, then the ball will hit the ground at the terminal velocity of 147.44 feet per second.
To do part A I will need to do a bit of counting.

[Guest]

I reckon the ball hits the ground at 9.9 seconds, travelling at 143.57 feet per second, which means it did not quite reach terminal velocity.
I got as far as expressions for velocity and distance travelled (involving tanh and cosh etc), got lost trying to extract "t" from the expressions so put the expressions in a spreadsheet and calculated by building a table for the range of time to fall.

[Guest]

I'm new to this forum stuff... I attached a pdf of 1 possible solution.

[Jeff3355]

I forgot to log in... I posted a pdf in other post above.
All you guys are correct! Nice!

[Guest]

Yes, good result, well presented and easy to follow...........I simply just got lost earlier