How do I solve this calculus problem regarding drag force and gravity?

Answer 1

I'm having trouble finding the relation between velocity and time given this formula. If anyone has a solve for this, It would be great if you took your time to help me as I know this includes differential acceleration, which requires calculus. As velocity increases, drag force increases - until terminal velocity.

F=1/2 * p * v^2 * C * A

p is density of fluid and is set to 1.225 kg/m^3

v is velocity of object relative to fluid

C is Coefficient of Drag and is set to 0.9, as my shape is a sphere

A is cross-sectional area which is π/4, as it is the "CS" of a sphere where diameter of "CS" circle in sphere is 1.

From my knowledge, I was only able to proceed this far:

Lets assume 1/2 * p * v^2 * C * A=ma1

Lets assume k=1/2pC (A is not included because I'll be comparing graphs with different values for A)

F=ma is the Net Force

g=9.80665m/s/s

F=mg

mg-ma1=ma

Force of gravity-Force of Drag=Net Force.

mg - 1/2 * k * v^2 * A = ma

g - (1/2 * k * v^2 * A) / m = a

How do I proceed? Do I differentiate a for dv/dt and integrate it later? I suck at calculus and I don't understand much of its logic, please help. If you have extra time, please give me a graph for velocity given time and acceleration given time.