Unizor - Physics4Teens - Mechanics - Dynamics - Free Falling
Channel: Zor Shekhtman on UNIZOR Education
Date: 2018-07-10

Unizor - Physics through a Looking Glass of Mathematician

Notes to a video lecture on http://www.unizor.com

Free Falling

Free falling is a movement of an object on a surface of a planet relative to this surface, when the only force acting on this object is the gravitational force of a planet.

In this case the one and only force of attraction acting on an object and directed towards the center of a planet can be expressed using the Law of Gravitation as follows:
F = G·M·m /R²
where G is a gravitational constant,
G = 6.674·10−11 N·m²/kg²

Knowing the force of gravity F and mass of an object m, we can determine the acceleration using the Newton's Second Law:
a = F/m = G·M /R²

Notice that this acceleration does not depend on m - mass of an object, which means that all objects fall on the surface of a planet with the same acceleration.
An interesting aspect of this formula is that we can imagine how to measure an acceleration (easy) and radius of a planet (more difficult, but possible), while we have no idea how to measure the mass of a planet.
So, this formula is used exactly for this purpose - to determine the mass of a planet, resolving the formula above for M:
M = a·R² /G

Let's solve a different problem now. We'd like to launch a satellite around the Earth that circulates around the planet at height H. What linear speed should a satellite have to stay on a circular orbit?

We know from Kinematics that an object rotating along a circular trajectory of radius r and angular speed ω has acceleration a=r·ω².
In terms of linear speed V=r·ω along an orbit this formula looks like
a = V²/r

Since the radius of an orbit is the radius of Earth R plus the height above its surface H, we should replace r in this formula with R+H.

The force of gravity is the only force acting on a satellite and the only source of its acceleration towards the Earth, so the acceleration above must be equal to acceleration of a free fall of a satellite. Here we will take into consideration already known mass of Earth and use distance from the center of the Earth to satellite as R+H, where R is the radius of Earth and H is a height above the Earth's surface.

The acceleration of a free fall to Earth at height H above the surface, using its radius R and already calculated mass of Earth M, is:
a = G·M/(R+H)²

Therefore, equating the acceleration of free fall to acceleration of an object rotating along a circular orbit, we come to the following equation:
V²/(R+H) = G·M/(R+H)²
from which we derive the value of required linear speed V:
V² = G·M/(R+H)