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The force of gravity for objects
relatively close to Earth equals the mass of the object
times the acceleration due to gravity. From this simple
equation, we can determine what happens when an object
is dropped from a height. We can also calculate what
happens when an object is propelled horizontal or
upward.
Questions you may have are:
-
What is the simple gravity equation?
-
What are other gravity equations on
Earth?
-
When happens when you propel an
object upwards?
Simple gravity equation
When you drop an object from a height
that is relatively close to the Earth, the gravitational
force pulling it to the ground is:
F = m*g
where
-
F is the force of gravity on
an object
-
m is the mass of the falling
object
-
g is the acceleration due to
gravity on Earth
The value of g is 32 ft/s2 (feet per
second squared) or 9.8 m/s2 (meters per second squared).
The units are sometimes written ft/s/s or m/s/s.
Since g is a constant, it means that all
objects fall to the earth at the same rate of
acceleration, no matter how much they weigh.
Relatively close to Earth means not so
far up that you are entering outer space.
Gravity equations
From the equation F = m*g, you can
determine the relationships between velocity, distance
and time for a falling object.
Force is defined as mass times
acceleration F = m*a. Thus m*a = m*g and a = g. That
means that the acceleration due to gravity is
independent of the mass. It also applies to the
velocity, distance and time it takes an object to fall.
They are all independent of the mass of the object.
It looks so simple, but many people
don't understand that all objects fall at the same rate,
independent of their mass, assuming air resistance is
negligible.
Since acceleration is the change in
velocity over time and since g is a constant, the
velocity v of a falling object is:
v = g*t
where t is the time in seconds after it
is dropped.
In other words, if an object drops for 2
seconds, its velocity will be v = 32 ft/s2 * 2 s = 64
ft/sec.
Also, since t = v/g, if an object is
falling at a rate of 29.4 meters/sec, it has been
falling for t = 29.4 m/s divided by 9.8 m/s/s = 3
seconds.
Since velocity is the change in velocity
over time and since v = g*t, integration results in:
x = g*t2/2
where x is the distance an object falls
over a time t.
For example, an object falling for t = 3
seconds would travel x = 32 ft/s2 * 9 s2 / 2 = 144 feet.
(Note how the units multiply together: ft/s2 * s2 = ft.)
Rearranging the distance equation, you
can calculate the time it takes an object to fall from a
height of x.
x = g*t2/2
2*x = g*t2
2*x/g = t2
thus
t = SQRT(2*x/g) or t = √(2*x/g)
where SQRT and √ mean the square root of
the quantity in the parentheses.
If you dropped a weight from a height of
64 feet, you could calculate how long it takes to hit
the ground.
t = √(2 * 64 ft / 32 ft/s2) = √4 = 2
seconds
Substituting t = 2 seconds into v = g*t,
you get a velocity of 64 feet/second.
Propelling an object
An object moving straight up from the
Earth will be slowed down due to the Force of gravity.
If the object is moving parallel to the Earth's surface,
its motion in that direction will not be slowed, but it
will drop at the same rate as if it were standing still.
If you drop an object from a given
height, it starts at v = 0 and accelerates due to the
force of gravity until it hits the ground at some new
velocity. Likewise, if you throw a ball upwards, the
velocity required to reach a given height would be the
same as the velocity the ball would achieve if it fell
from that height.
Going straight up is a mirror image of
going straight down. This is because of the Law of
Conservation of Energy.
You can calculate how high the ball will
go when thrown at a given velocity with the equation x =
v2/2*g, where x is the height, v2 is the velocity times
itself or velocity squared, and g is the gravitational
acceleration of 32 ft/s2 or 9.8 m/s2.
Thus, if you throw a ball up at v = 9.8
m/s, it would go to a height of x = 4.9 m.
If you would shoot a bullet from a gun
exactly parallel to the Earth's surface, the motion of
the bullet would have no effect on how gravity acts on
the bullet. In other words, the bullet would drop at the
same rate as a stationary object.
Many people don't believe that if you
held a rifle parallel to the ground and at the same time
you shot the bullet, you dropped another bullet from the
same height, that they would both hit the ground at the
same time. This would be true even if the bullet shot
from the gun traveled a great distance before hitting
the ground.
You can try a simple experiment to
verify this. Place a coin on the edge of a table or desk
and hold another coin at the same height. With one hand
flip the coin on the table across the room. At the same
time, drop the other coin. You will or hear that they
hit the floor at just about the same time.
Both throwing a basketball to a hoop and
lobbing a cannon shell are propelling the object at an
upward angle. The object will follow a parabolic arc.
This is a combination of shooting straight up and
shooting parallel to the ground. The two components
result in the curved path.
Calculating that path is beyond what we
will cover in this lesson.
The force of gravity close to Earth
equals the mass of the object times the acceleration due
to gravity. From this simple equation, we can determine
what happens when an object is dropped from a height. We
can also calculate what happens when an object is
propelled horizontal or upward. |
