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Viscous force and gravitation

Storyboard

When an object moves through a viscous medium under the influence of a constant force like gravity, initially gravity accelerates the object until its velocity increases to a level where the viscous force and gravity balance out. From this point onward, the object no longer accelerates and moves at a constant velocity.

>Model

ID:(1965, 0)



Mechanisms

Iframe

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Code
Concept
Viscous force

Mechanisms

Fall pathFall speedForces on a sphereOstwald methodViscous force

ID:(15539, 0)



Viscous force on a body

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The force experienced by a body moving at ($$) in a medium, characterized by the constant of the Viscose Force (b), is the viscose force (F_v), as described by the equation:

F_v = b v



To understand the role of the constant of the Viscose Force (b), it's important to remember that viscosity is a measure of how momentum, or the velocity of molecules, diffuses. In other words, the constant of the Viscose Force (b) represents the extent to which the body loses energy by transferring it to the medium and accelerating the molecules, thereby providing them with energy. Therefore, the constant of the Viscose Force (b) is proportional to viscosity.

ID:(15546, 0)



Forces on a sphere falling in a medium

Description

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When a sphere is thrown into a viscous medium, there's an initial upward force, a gravitational Force (F_g), which gradually sinks the body. During this process, the sphere gains velocity, resulting in a downward force, a viscose force (F_v), which depends on the velocity. As the total velocity, the force with constant mass (F),

F = F_g - F_v



begins to decrease until it becomes null. From this moment on, the movement continues at a constant velocity since there's no force to accelerate it.

ID:(15544, 0)



Ostwald method for measuring viscosity

Description

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The Ostwald viscosity measurement method is based on the behavior of a liquid flowing through a small-radius tube (capillary).

The liquid is introduced, suction is applied to exceed the upper mark, and then it is allowed to drain, measuring the time it takes for the level to pass from the upper to the lower mark.

The experiment is conducted first with a liquid for which viscosity and density are known (e.g., distilled water), and then with the liquid for which viscosity is to be determined. If conditions are identical, the liquid flowing in both cases will be similar, and thus, the time will be proportional to the density divided by the viscosity. Thus, a comparison equation between both viscosities can be established:

ID:(15545, 0)



Fall speed in viscous medium

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In the case of a body falling in a viscous medium, the equation of motion is an equation of the speed (v) as a function of the time (t) with the gravitational mass (m_g), the inertial Mass (m_i), the gravitational Acceleration (g), and the constant of the Viscose Force (b):

m_i \displaystyle\frac{dv}{dt} = m_g g - b v



This is obtained with the viscosity time and inertial mass (\tau_i)

\tau_i \equiv \displaystyle\frac{ m_i }{ b }



and with the viscosity time and gravitational mass (\tau_g)

\tau_g \equiv \displaystyle\frac{ m_g }{ b }



Integrating with initial time zero and the initial Speed (v_0),

v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



which is represented below:



The graph illustrates how viscosity compels the body to descend with ($$), which equals g\tau_g. This occurs at a time of the order of the viscosity time and inertial mass (\tau_i), whether the speed (v) is less than or greater than the asymptotic speed (v_{\infty}).

ID:(15547, 0)



Fall path in viscous medium

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In the case of a body falling in a viscous medium, the equation of motion is an equation of the position (s) as a function of the gravitational Acceleration (g), the viscosity time and inertial mass (\tau_i), the viscosity time and gravitational mass (\tau_g), the initial Speed (v_0), and the time (t):

\displaystyle\frac{ds}{dt} = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



From this equation, we obtain by integrating with initial time zero and ($$):

s = s_0 + g \tau_g t +( v_0 - g \tau_g ) \tau_i (1 - e^{- t / \tau_i })



which is represented below:

ID:(15550, 0)



Model

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Parameters

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
v_{\infty}
v_inf
Asymptotic speed
m/s
b
b
Constant of the Viscose Force
kg/s
g
g
Gravitational Acceleration
m/s^2
m_g
m_g
Gravitational mass
kg
m_i
m_i
Inertial Mass
kg
v_0
v_0
Initial Speed
m/s
s_0
s_0
Starting position
m
\tau_g
tau_g
Viscosity time and gravitational mass
s
\tau_i
tau_i
Viscosity time and inertial mass
s

Variables

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
F
F
Force with constant mass
N
F_g
F_g
Gravitational Force
N
a
a
Instant acceleration
m/s^2
s
s
Position
m
v
v
Speed
m/s
t
t
Time
s
F_v
F_v
Viscose force
N

Calculations


First, select the equation: to , then, select the variable: to
@DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) F = F_g - F_v F = m_i * a F_g = m_g * g F_v = b * v m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v m_i * a = m_g * g - b * v s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) tau_g = m_g / b tau_i = m_i / b v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_gv_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

Calculations

Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

Variable Given Calculate Target : Equation To be used
@DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) F = F_g - F_v F = m_i * a F_g = m_g * g F_v = b * v m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v m_i * a = m_g * g - b * v s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) tau_g = m_g / b tau_i = m_i / b v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_gv_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i




Equations

#
Equation

\displaystyle\frac{ds}{dt} = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }

@DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i )


F = F_g - F_v

F = F_g - F_v


F = m_i a

F = m_i * a


F_g = m_g g

F_g = m_g * g


F_v = b v

F_v = b * v


m_g = m_i

m_g = m_i


m_i \displaystyle\frac{dv}{dt} = m_g g - b v

m_i * @DIFF( v , t ) = m_g * g - b * v


m_i a = m_g g - b v

m_i * a = m_g * g - b * v


s = s_0 + g \tau_g t +( v_0 - g \tau_g ) \tau_i (1 - e^{- t / \tau_i })

s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i ))


\tau_g \equiv \displaystyle\frac{ m_g }{ b }

tau_g = m_g / b


\tau_i \equiv \displaystyle\frac{ m_i }{ b }

tau_i = m_i / b


v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }

v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i )


v_{\infty} \equiv g \tau_g

v_i = g * tau_g

ID:(15541, 0)



Total force of body falling in viscous medium

Equation

>Top, >Model


In the case of a body falling in a viscous medium, the total force, the force with constant mass (F), is equal to the gravitational Force (F_g) minus the viscose force (F_v), so

F = F_g - F_v

F
Force with constant mass
N
9046
F_g
Gravitational Force
N
4977
F_v
Viscose force
N
4979
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(15543, 0)



Force case constant mass

Equation

>Top, >Model


In the case where the inertial Mass (m_i) equals the initial mass (m_0),

m_g = m_i



the derivative of momentum will be equal to the mass multiplied by the derivative of the speed (v). Since the derivative of velocity is the instant acceleration (a), we have that the force with constant mass (F) is

F = m_i a

F
Force with constant mass
N
9046
m_i
Inertial Mass
kg
6290
a
Instant acceleration
m/s^2
4972
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

Since the moment (p) is defined with the inertial Mass (m_i) and the speed (v),

p = m_i v



If the inertial Mass (m_i) is equal to the initial mass (m_0), then we can derive the momentum with respect to time and obtain the force with constant mass (F):

F=\displaystyle\frac{d}{dt}p=m_i\displaystyle\frac{d}{dt}v=m_ia



Therefore, we conclude that

F = m_i a

ID:(10975, 0)



Viscose Force

Equation

>Top, >Model


The simplest form of the viscose force (F_v) is one that is proportional to the the speed (v) of the body, represented by:

F_v = b v

b
Constant of the Viscose Force
kg/s
5312
v
Speed
m/s
6029
F_v
Viscose force
N
4979
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i



The proportionality constant, also known as the constant of the Viscose Force (b), generally depends on the shape of the object and the viscosity of the medium through which it moves. An example of this type of force is the one exerted by a fluid stream on a spherical body, whose mathematical expression is known as Stokes' law.

ID:(3243, 0)



Gravitational Force

Equation

>Top, >Model


The gravitational Force (F_g) is based on the gravitational mass (m_g) of the object and on a constant reflecting the intensity of gravity at the planet's surface. The latter is identified by the gravitational Acceleration (g), which is equal to 9.8 m/s^2.

Consequently, it is concluded that:

F_g = m_g g

g
Gravitational Acceleration
9.8
m/s^2
5310
F_g
Gravitational Force
N
4977
m_g
Gravitational mass
kg
8762
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(3241, 0)



Equality of inertial and gravitational mass

Equation

>Top, >Model


The masses that Newton used in his principles are related to the inertia of bodies, which leads to the concept of the inertial Mass (m_i).

Newton's law, which is linked to the force between bodies due to their masses, is related to gravity, hence known as the gravitational mass (m_g).

Empirically, it has been concluded that both masses are equivalent, and therefore, we define

m_g = m_i

m_g
Gravitational mass
kg
8762
m_i
Inertial Mass
kg
6290
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

Einstein was the one who questioned this equality and, from that doubt, understood why both 'appear' equal in his theory of gravity. In his argument, Einstein explained that masses deform space, and this deformation of space causes a change in the behavior of bodies. Thus, masses turn out to be equivalent. The revolutionary concept of space curvature implies that even light, which lacks mass, is affected by celestial bodies, contradicting Newton's theory of gravitation. This was experimentally demonstrated by studying the behavior of light during a solar eclipse. In this situation, light beams are deflected due to the presence of the sun, allowing stars behind it to be observed.

ID:(12552, 0)



Equation of motion falling in a viscous medium

Equation

>Top, >Model


The force with constant mass (F) equals the gravitational Force (F_g) minus the viscose force (F_v) so that:

F = F_g - F_v



This relation allows establishing the motion equation for the instant acceleration (a) with ($$) falling due to Earth's gravity with the gravitational Acceleration (g), and with ($$), in the constant of the Viscose Force (b), it will take the form of:

m_i a = m_g g - b v

b
Constant of the Viscose Force
kg/s
5312
g
Gravitational Acceleration
9.8
m/s^2
5310
m_g
Gravitational mass
kg
8762
m_i
Inertial Mass
kg
6290
a
Instant acceleration
m/s^2
4972
v
Speed
m/s
6029
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(14495, 0)



Differential equation for the fall in a viscous medium

Equation

>Top, >Model


The inertial Mass (m_i) falling due to Earth's gravity with ($$), and with ($$) in a viscous medium with ($$), is presented as follows:

m_i a = m_g g - b v



To solve this equation, it is necessary to bring it to its differential form. This is achieved by replacing the instant acceleration (a) with the derivative of the speed (v) in the time (t):

m_i \displaystyle\frac{dv}{dt} = m_g g - b v

b
Constant of the Viscose Force
kg/s
5312
g
Gravitational Acceleration
9.8
m/s^2
5310
m_g
Gravitational mass
kg
8762
m_i
Inertial Mass
kg
6290
v
Speed
m/s
6029
t
Time
s
5264
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(14492, 0)



Gravitational mass time and viscosity

Equation

>Top, >Model


With the equation of motion of a body in a viscous medium, we have the derivative of the speed (v) at the time (t) with the inertial Mass (m_i), the gravitational mass (m_g), the constant of the Viscose Force (b) and the gravitational Acceleration (g):

m_i \displaystyle\frac{dv}{dt} = - b v



This defines the viscosity time and gravitational mass (\tau_g) as:

\tau_g \equiv \displaystyle\frac{ m_g }{ b }

b
Constant of the Viscose Force
kg/s
5312
m_g
Gravitational mass
kg
8762
\tau_g
Viscosity time and gravitational mass
s
10329
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(15549, 0)



Inertial mass time and viscosity

Equation

>Top, >Model


With the equation of motion of a body in a viscous medium, we have the derivative of the speed (v) at the time (t) with the constant of the Viscose Force (b) and the gravitational Acceleration (g):

m_i \displaystyle\frac{dv}{dt} = - b v



This defines the viscosity time and inertial mass (\tau_i) as:

\tau_i \equiv \displaystyle\frac{ m_i }{ b }

b
Constant of the Viscose Force
kg/s
5312
m_i
Inertial Mass
kg
6290
\tau_i
Viscosity time and inertial mass
s
10328
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(15548, 0)



Solution of motion falling in a viscous medium

Equation

>Top, >Model


The equation of motion in the speed (v) at the time (t) with the inertial Mass (m_i), the gravitational mass (m_g), the gravitational Acceleration (g), and the constant of the Viscose Force (b):

m_i \displaystyle\frac{dv}{dt} = m_g g - b v



Assuming the initial time is zero, the initial Speed (v_0), the viscosity time and gravitational mass (\tau_g), and the viscosity time and inertial mass (\tau_i), we obtain the following equation:

v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }

g
Gravitational Acceleration
9.8
m/s^2
5310
v_0
Initial Speed
m/s
5188
v
Speed
m/s
6029
t
Time
s
5264
\tau_g
Viscosity time and gravitational mass
s
10329
\tau_i
Viscosity time and inertial mass
s
10328
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

The equation of motion at the speed (v) in the time (t) with the inertial Mass (m_i), the gravitational mass (m_g), the gravitational Acceleration (g), and the constant of the Viscose Force (b):

m_i \displaystyle\frac{dv}{dt} = m_g g - b v



along with the definition of the viscosity time and inertial mass (\tau_i)

\tau_i \equiv \displaystyle\frac{ m_i }{ b }



and the viscosity time and gravitational mass (\tau_g)

\tau_g \equiv \displaystyle\frac{ m_g }{ b }



can be reformulated as

\displaystyle\frac{dv}{g\tau_g - v} = \displaystyle\frac{dt}{\tau_i}



If we integrate this expression between ($$) and the speed (v), and from the initial time zero until the time (t), we obtain

\ln(g\tau_g-v_0)-\ln(g\tau_g-v)=\displaystyle\frac{t}{\tau_i}



Solving for velocity, we get

v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



This equation illustrates that the initial Speed (v_0) then asymptotically converges to the velocity g\tau_g.

ID:(14493, 0)



Asymptotic speed

Equation

>Top, >Model


The integration of the equation of motion yields the speed (v) as a function of the gravitational Acceleration (g), the viscosity time and inertial mass (\tau_i), the viscosity time and gravitational mass (\tau_g), the initial Speed (v_0) and the time (t) in the form:

v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



For the time (t) much larger than the viscosity time and inertial mass (\tau_i), the limit the asymptotic speed (v_{\infty}) is obtained:

v_{\infty} \equiv g \tau_g

v_{\infty}
Asymptotic speed
m/s
10070
g
Gravitational Acceleration
9.8
m/s^2
5310
\tau_g
Viscosity time and gravitational mass
s
10329
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(14494, 0)



Differential equation falling path in viscous medium

Equation

>Top, >Model


The integration of the equation of motion yields the speed (v) as a function of the gravitational Acceleration (g), the viscosity time and inertial mass (\tau_i), the viscosity time and gravitational mass (\tau_g), the initial Speed (v_0) and the time (t) in the form:

v = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



in its differential form,

\displaystyle\frac{ds}{dt} = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }

g
Gravitational Acceleration
9.8
m/s^2
5310
v_0
Initial Speed
m/s
5188
s
Position
m
9899
t
Time
s
5264
\tau_g
Viscosity time and gravitational mass
s
10329
\tau_i
Viscosity time and inertial mass
s
10328
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i



where the position (s) represents the distance traveled.

ID:(14496, 0)



Path traveled in a fall in a viscous medium

Equation

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The integration of the equation of motion yields the position (s) in terms of the gravitational Acceleration (g), the viscosity time and inertial mass (\tau_i), the viscosity time and gravitational mass (\tau_g), the initial Speed (v_0) and the time (t) as follows:

\displaystyle\frac{ds}{dt} = g \tau_g + ( v_0 - g \tau_g )e^{- t / \tau_i }



from an initial null time to the time (t), and from the starting position (s_0) to the position (s), we obtain

s = s_0 + g \tau_g t +( v_0 - g \tau_g ) \tau_i (1 - e^{- t / \tau_i })

g
Gravitational Acceleration
9.8
m/s^2
5310
v_0
Initial Speed
m/s
5188
s
Position
m
9899
s_0
Starting position
m
5336
t
Time
s
5264
\tau_g
Viscosity time and gravitational mass
s
10329
\tau_i
Viscosity time and inertial mass
s
10328
F_g = m_g * g F_v = b * v F = m_i * a m_g = m_i m_i * @DIFF( v , t ) = m_g * g - b * v v = g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) v_i = g * tau_g m_i * a = m_g * g - b * v @DIFF( s , t )= g * tau_g + ( v_0 - g * tau_g )*exp(- t / tau_i ) s = s_0 + g * tau_g * t + ( v_0 - g * tau_g )* tau_i *(1-exp(- t / tau_i )) F = F_g - F_v tau_i = m_i / b tau_g = m_g / b v_infbFgF_gm_gm_iv_0asvs_0tF_vtau_gtau_i

ID:(14497, 0)