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Gravitational force
Storyboard 
Gravitational force is defined as the gravitational mass multiplied by the gravitational acceleration.The gravitational acceleration depends on the planet or moon being considered. While on Earth, the gravitational acceleration $g$ is 9.8 m/s², on the Moon, it's 1.625 m/s².
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Inertial and gravitational mass equality
Description
The gravitational mass
This was shown by the astronauts in Apollo 15. The first part contains the original video, the second a Hollywood type version.
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Gravitational force
Model
Gravitational force is defined as the gravitational mass multiplied by the gravitational acceleration. The gravitational acceleration depends on the planet or moon being considered. While on Earth, the gravitational acceleration $g$ is 9.8 m/s², on the Moon, it's 1.625 m/s².
Variables
Calculations
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Calculations
Variable
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Equation
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Equations
In the case where the constant Acceleration ($a_0$) equals the mean Acceleration ($\bar{a}$), it will be equal to
| $ a_0 = \bar{a} $ |
.
Therefore, considering the speed Diference ($\Delta v$) as
| $ dv \equiv v - v_0 $ |
and the time elapsed ($\Delta t$) as
| $ \Delta t \equiv t - t_0 $ |
,
the equation for the constant Acceleration ($a_0$)
| $ \bar{a} \equiv\displaystyle\frac{ \Delta v }{ \Delta t }$ |
can be written as
$a_0 = \bar{a} = \displaystyle\frac{\Delta v}{\Delta t} = \displaystyle\frac{v - v_0}{t - t_0}$
and by rearranging, we obtain
| $ v = v_0 + a_0 ( t - t_0 )$ |
.
(ID 3156)
In the case of the constant Acceleration ($a_0$), the speed ($v$) as a function of the time ($t$) forms a straight line passing through the start Time ($t_0$) and the initial Speed ($v_0$), defined by the equation:
| $ v = v_0 + a_0 ( t - t_0 )$ |
Since the distance traveled in a time ($\Delta s$) represents the area under the velocity-time curve, we can sum the contributions of the rectangle:
$v_0(t-t_0)$
and the triangle:
$\displaystyle\frac{1}{2}a_0(t-t_0)^2$
To obtain the distance traveled in a time ($\Delta s$) with the position ($s$) and the starting position ($s_0$), resulting in:
| $ \Delta s = s - s_0 $ |
Therefore:
| $ s = s_0 + v_0 ( t - t_0 )+\displaystyle\frac{1}{2} a_0 ( t - t_0 )^2$ |
(ID 3157)
If we solve for the time ($t$) and the start Time ($t_0$) in the equation of the speed ($v$), which depends on the initial Speed ($v_0$) and the constant Acceleration ($a_0$):
| $ v = v_0 + a_0 ( t - t_0 )$ |
we get:
$t - t_0= \displaystyle\frac{v - v_0}{a_0}$
And when we substitute this into the equation of the position ($s$) with the starting position ($s_0$):
| $ s = s_0 + v_0 ( t - t_0 )+\displaystyle\frac{1}{2} a_0 ( t - t_0 )^2$ |
we obtain an expression for the distance traveled as a function of velocity:
| $ s = s_0 +\displaystyle\frac{ v ^2- v_0 ^2}{2 a_0 }$ |
(ID 3158)
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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 $ |
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Examples
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(ID 15417)
Gravitational mass is associated with what Newton defined as the law of gravitation, indicating the force that one body exerts on another.It should not be confused with inertial mass, which indicates the resistance a body generates when changing its state of motion. The latter is associated with the inertia experienced by bodies and is referred to as inertial mass.
(ID 14464)
The gravitational mass
This was shown by the astronauts in Apollo 15. The first part contains the original video, the second a Hollywood type version.
(ID 11026)
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