Resource Lesson
Freefall: Projectiles in 1-Dimension
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A
projectile
is an object that moves through the air by virtue of its own inertia. Recall that mass is a measure of an object's inertia which is its resistance to a change in its state of motion. The term
freefall
means that the only force acting on the projectile is gravity; that is, there is no air resistance present. While in freefall, all projectiles experiences a unique value for their vertical acceleration: a = -g = -9.81 m/sec
^{2}
. The term
trajectory
means the projectile’s path through the air. If the projectile only has vertical velocity, its trajectory traces out a vertical line. When it has a constant horizontal velocity combined with a vertical velocity which is uniformly accelerated, the trajectory will be parabolic. The term
apex
means highest point in the projectile’s trajectory where its instantaneous vertical velocity equals 0.
Refer to the following information for the next four questions.
Take a moment to remember the kinematics equations for uniformly accelerated motion.
What kinematics formula relates the variables s, v
_{o}
, a and t ?
What kinematics formula relates v
_{f}
, v
_{o}
, a, and s?
What kinematics formula relates v
_{f}
, v
_{o}
, a, and t?
What kinematics formula relates s, v
_{o}
, v
_{f}
and t?
These formulas are used when the
acceleration is uniform (constant)
. Remember that acceleration is the rate of change of velocity. If the object is either losing speed while traveling in a positive direction OR gaining speed in a negative direction, a is negative.
In freefall problems, a has a value of -9.81 m/sec
^{2}
.
This value is represented by the variable
g
. Which is either called the "acceleration due to gravity" or the "gravitational field strength." Its value depends on where the projectile is located with respect to the center of the earth. The value for
g
on the surface of the earth is derived based on the formula for universal gravitation and weight.
weight = force of universal gravitation
|mg| = G(mM
_{E}
/r
^{2}
)
|g| = G(M
_{E}
/r
^{2}
)
Try it! G = 6.67 x 10
^{-11}
Nm
^{2}
/kg
^{2}
, M
_{E}
= 5.98 x 10
^{24}
kg, r = R
_{E}
= 6.37 x 10
^{6}
m
Substituting these values give us the magnitude of
g
to be approximately 9.8 m/sec
^{2}
. Since gravity pulls objects towards the center of the earth, the value of
a
used in our kinematics equations for uniformly accelerated motion when working freefall problems will be
a = - g = -9.8 m/sec
^{2}
.
Note that only the vertical motion of the projectile will experience this acceleration since gravity is a "vertical force" that attracts the projectile to the "center of the earth." For this reason, we often subscript the variable as
a
_{y}
= - 9.8 m/sec
^{2}
. Any projectile moving in two-dimensions will experience no acceleration horizontally since freefall eliminates all forces (air resistance, drag) except for the pull of gravity.
The remainder of this lesson only deals with one-dimensional, or vertical freely falling bodies, so we will just use the notation a = -9.8 m/sec
^{2}
.
For a projectile thrown vertically straight upwards, examine the sketch below which relates the graphs for the projectile's
position vs time
and its
velocity vs time
.
When doing freefall /projectile problems,
vertical velocities, v
, are
positive
when the object is traveling "up" towards the apex, and
negative
when the object is falling "down" after having reached the apex.
While the
displacement , s
, is
positive when the projectile is located "above the release height,"
negative when located "below the release height," and
equal to zero when the projectile has returned to its original release height.
Refer to the following information for the next five questions.
Identify the positions (A, B, C, D, or E) that represent each set of criteria.
v
_{f}
> 0 and s > 0
v
_{f}
< 0 and s < 0
v
_{f}
< 0 and s > 0
v
_{f}
= 0 and s > 0
v
_{f}
< 0 and s = 0
Now let's apply our knowledge to some problems that contain numerical data. In each of the following scenarios, state the values of v
_{o}
, a, and s.
A rock dropped from a 20 meter bridge falls into the river below.
A rock thrown upwards at 6 m/sec from a 20 meter bridge falls into the river below.
A rock thrown downwards at 6 m/sec from a 20 meter bridge falls into the river below.
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