# Differential equation for grav, boyant and drag force

• Agustin
In summary, the conversation discusses the calculation of a time-dependent distance equation for a falling object, taking into account the boyant force, gravity, and drag force. The equation involves variables such as drag coefficient, area, air density, and velocity. The final equation is dimensionally correct and can be further tested for accuracy in different scenarios.
Agustin

## Homework Statement

So there is a falling object, you have to take into account the boyant force, the pull of gravity and the drag force
A time dependent distance equation is what we're looking for

## Homework Equations

Fd=CdApav2/2
Where
Fd is the drag force
Cd is the drag coefficient
A is the area exposed to the fall
pa the air density
v the immediate velocity

Fb=mgpa/pc
Fb is the boyant force
mg is the weight of the object
pc is the object's density
Note ( this equation is found from the original equation Fb=Volume submerged x air density x gravity; where the submerged volume is m/pc)

Fg=mg

## The Attempt at a Solution

ma=mg - mgpa/pc - CdApav2/2
Or
dv/dt = A - Bv2
Where
A=g( 1 - pa/pc)
B=CdApa/2m

I solve for v

v (t) = (A/B)1/2 (1 + C1e-2(BA)1/2t)/( 1 - C1e-2(BA)1/2t)

Where C1 is some arbitrary constant

Integrating we get the distance formula:

X (t) = (A/B)1/2t+(1/B)ln( 1 - C1e-2 (BA)1/2t) + C2
I don't know wether it's correct or not. I've used techniques i found on the internet for the integration. http://www.freemathhelp.com/forum/threads/47073-integral-(1-(e-x-1))-dx-Using-Partial-Fractions

See if it gives the right answers in special cases. For example, what does your result say for the case of no buoyant force and no drag force? What happens as ## t \rightarrow \infty ##, and what would you expect physically in this case? There are lots of things like this to look into.

Oh, and can you check that the resulting equation is dimensionally correct?

## What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is commonly used to model physical systems and describe how they change over time.

## How is gravity incorporated into differential equations?

Gravity is incorporated into differential equations through the use of the gravitational force, which is represented by the equation F = mg. This force is then used in the differential equation to describe the motion of a body under the influence of gravity.

## What is the role of buoyancy in differential equations?

Buoyancy is an upward force that acts on an object when it is immersed in a fluid. In differential equations, buoyancy is taken into account by using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.

## How is drag force included in differential equations?

Drag force is incorporated into differential equations by using the drag equation, which takes into account factors such as the speed, density, and shape of an object. This equation is used to describe how air resistance affects the motion of an object.

## Can differential equations accurately model real-world systems?

While differential equations can be used to model physical systems, they are not always able to accurately predict real-world behavior. Factors such as external forces, friction, and other complex variables can make it challenging to create a completely accurate model.

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