# I need the name of this equation for my IA?

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Hi guys, I found this equation on 2 other forums, and I'm writing a Math IA with Physics formulas integrated into it. I would really appreciate it if you could give me the name of this formula, or a method to derive it (I don't have a strong physics background) so that I could cite it in my paper or refer to it.

m(d2x/dt2) = (rhoB-rhoA)(g*V) - 0.5(Cd)(rhoA)(A)(dx/dt)^2

Where m = body's mass, rhoA = fluid's density, rhoB = body's density, g = gravity (may be assumed to vary with height -> which is where the "effects of gravity" come in), x = displacement, t = time, Cd = drag coefficient, V = body volume, A = body area.

^I found this equation on this forum: https://www.physicsforums.com/threads/gravity-in-water.6463/

*Some extra information* : For my Paper, im trying to represent a mathematical model for a swimming race dive for both above and in the water. For above the water, I am using a simple height formula "h(t)=1/2at^2+vt+h". But as there are many other factors such as drag and buoyancy underwater, I need to find a coherent formula with the same variable of time for height underwater.

Any help would be really appreciated!! :)

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256bits
Gold Member
m(d2x/dt2) = (rhoB-rhoA)(g*V) - 0.5(Cd)(rhoA)(A)(dx/dt)^2
Do you know what the terms in the equation mean?
Hints:
m(d2x/dt2) is the net ___________ acting on the object.

(rhoB-rhoA)(g*V) is the net force acting on the object in a fluid.
Which comprises the ________of the object due to gravity, and the ___________ force acting due to the object being immersed in a fluid.

- 0.5(Cd)(rhoA)(A)(dx/dt)^2 is the ____________ ____________ of an object travelling through a fluid.

http://hyperphysics.phy-astr.gsu.edu/hbase/mass.html
https://en.wikipedia.org/wiki/Buoyancy
https://en.wikipedia.org/wiki/Drag_equation
https://en.wikipedia.org/wiki/Terminal_velocity

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Do you know what the terms in the equation mean?
Hints:
m(d2x/dt2) is the net ___________ acting on the object.

(rhoB-rhoA)(g*V) is the net force acting on the object in a fluid.
Which comprises the ________of the object due to gravity, and the ___________ force acting due to the object being immersed in a fluid.

- 0.5(Cd)(rhoA)(A)(dx/dt)^2 is the ____________ ____________ of an object travelling through a fluid.

http://hyperphysics.phy-astr.gsu.edu/hbase/mass.html
https://en.wikipedia.org/wiki/Buoyancy
https://en.wikipedia.org/wiki/Drag_equation
https://en.wikipedia.org/wiki/Terminal_velocity
I think I understand what the equation stands for now! Thank you!!

But just to be safe, is this still applicable to finding a height (displacement) function for a fully submerged object if I were to integrate it to find x?

256bits
Gold Member
I think I understand what the equation stands for now! Thank you!!

But just to be safe, is this still applicable to finding a height (displacement) function for a fully submerged object if I were to integrate it to find x?
Well I don't have my math hat on right now, but if you are capable of doing the steps, go for it.
But the equation shows that for each dx the object moves, the force on the object changes, which will change the acceleration and thus the change in velocity will not be the same for each step. The suvat equations assume a constant acceleration.
You have a dx2/dt2, a dx/dt and a ( dx/dt ) ^2 in the equation.

I will have to step out for a bit.

Well I don't have my math hat on right now, but if you are capable of doing the steps, go for it.
But the equation shows that for each dx the object moves, the force on the object changes, which will change the acceleration and thus the change in velocity will not be the same for each step. The suvat equations assume a constant acceleration.
You have a dx2/dt2, a dx/dt and a ( dx/dt ) ^2 in the equation.

I will have to step out for a bit.
In any case you really helped me out, so thank you!!