# Sinking Object Motion Equations

• Johnnnnnnnn
In summary: Alternatively, you could use the Laplace equation to solve for y.I don't think the Laplace equation would be applicable here, as it's not in the form y'' + p(x)y' + q(x)y = g(x).I'm not sure what you're asking...Can you explain your question in more detail?In summary, the equation for velocity was derived based on trial and error.
Johnnnnnnnn
Hi guys! I am currently learning about fluid dynamics, and I am stuck on a certain equation derivation. It's about sinking motion which considers only gravity force, buoyant force, and viscous resistance. The link attached has the details.

http://hyperphysics.phy-astr.gsu.edu/hbase/lindrg.html#c2
The problem I have is how the equation for velocity was found. I understand that Fnet = mg - pVg - bv, which is basically net force = weight - buoyant force - viscous resistance, but I don't get how the velocity (equation below) was derived in terms of time, terminal velocity (Vt), and initial velocity (V0).

Is there a mathematical proof for this? Or was the equation created based on trial and error? If you could, could you also please explain how the equation for distance (image below) was also obtained? Thanks!

.

Hello Joh8n,

The Newton equation of motion ##F_{\rm net} = mg'- bv ## is a differential equation
(##F_{\rm net} = {dv\over dt} ##)

The solution is discussed here (6.4.10 onwards; they don't express in terms of ##v_0## and ##v_t##, but I hope you can follow the reasoning)

Johnnnnnnnn
Oh I see... Thanks for the help!

BvU said:
Hello Joh8n,

The Newton equation of motion ##F_{\rm net} = mg'- bv ## is a differential equation
(##F_{\rm net} = {dv\over dt} ##)

The solution is discussed here (6.4.10 onwards; they don't express in terms of ##v_0## and ##v_t##, but I hope you can follow the reasoning)

So after seeing the derivation of sinking motion, I was wondering if this method of derivation is also correct.

Since ##F_{\rm net} = ma = mg'- bv ##,
##a = g' - (b/m)v##

From here, could I set a = d2y/ dt2, and v = dy/dt, letting y be the vertical distance the object travels, and use second order linear differential equation to solve it?

So in another words, (using y' as the notation for dy/dt)

y'' = g - (b/m) y'
y'' + (b/m)y' = g

Would solving for y with this second order linear differential equation be a correct way to solve for y? If not, could you also please explain why such way is not applicable? Thanks!

It's still an equation in y', not in y ...

But I thought second order linear differential equations are in the form of y'' + p(x)y' + q(x)y = g(x), where q(x) would be 0 in this case. I'm guessing you mean that it's unnecessary to change v into y' as it just complicates the calculation?

Correct. You have an equation where y does not appear, but y' does. So you solve for y' and then integrate once to get y.

## 1. What is the equation for calculating the motion of a sinking object?

The equation for calculating the motion of a sinking object is d = 1/2 * g * t^2, where d is the distance the object has sunk, g is the acceleration due to gravity (9.8 m/s^2), and t is the time elapsed.

## 2. How does the density of an object affect its sinking motion?

The density of an object affects its sinking motion because objects with higher densities will sink faster than objects with lower densities. This is because the higher density objects have a greater mass, and therefore experience a greater force of gravity pulling them down.

## 3. What other factors can impact the sinking motion of an object?

Other factors that can impact the sinking motion of an object include the shape and size of the object, the viscosity of the fluid it is sinking in, and any external forces acting on the object, such as wind or currents.

## 4. How can the sinking motion of an object be graphically represented?

The sinking motion of an object can be graphically represented by plotting the distance the object has sunk over time. This will result in a parabolic curve, as the object's distance will increase at a faster rate as time goes on due to the acceleration of gravity.

## 5. Are there any real-life applications for sinking object motion equations?

Yes, there are many real-life applications for sinking object motion equations. For example, they can be used to predict the trajectory of a sinking ship or submarine, or to calculate the speed at which an object will hit the bottom of a body of water. They are also used in engineering and construction to design structures that will sink or float properly in different fluids.

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