It
has already been stated and thoroughly discussed that the
horizontal and vertical motions of a projectile are
independent of each other. The horizontal velocity of
a projectile does not affect how far (or how fast) a
projectile falls vertically. Perpendicular
components of motion are independent of each other.
Thus, an analysis of the motion of a projectile demands that
the two components of motion are analyzed independent of
each other, being careful not to mix horizontal
motion information with vertical motion information. That
is, if analyzing the motion to determine the vertical
displacement, one would use kinematic equations with
vertical motion parameters (initial vertical
velocity, final vertical velocity, vertical
acceleration) and not horizontal motion parameters (initial
horizontal velocity, final horizontal velocity, horizontal
acceleration). It is for this reason that one of the initial
steps of a projectile motion problem is to determine the
components of the initial velocity.

Earlier in this
unit, the method of vector resolution was discussed.
Vector resolution is the method of taking a single vector at
an angle and separating it into two perpendicular parts. The
two parts of a vector are known as components
and describe the influence of that vector in a single
direction. If a projectile is launched at an angle to the
horizontal, then the initial velocity of the projectile has
both a horizontal and a vertical component. The horizontal
velocity component
(v_{x})
describes the influence of the velocity in displacing the
projectile horizontally. The vertical velocity component
(v_{y})
describes the influence of the velocity in displacing the
projectile vertically. Thus, the analysis of projectile
motion problems begins by using the
trigonometric methods discussed earlier to determine the
horizontal and vertical components of the initial
velocity.

Consider a projectile launched with an
initial velocity of 50 m/s at an angle of 60 degrees above
the horizontal. Such a projectile begins its motion with a
horizontal velocity of 25 m/s and a vertical velocity of 43
m/s. These are known as the horizontal and vertical
components of the initial velocity. These numerical values
were determined by constructing a sketch of the velocity
vector with the given direction and then using trigonometric
functions to determine the sides of the velocity
triangle. The sketch is shown at the right and the use of
trigonometric functions to determine the magnitudes is shown
below. (If necessary, review this method on an
earlier page in this unit.)

All
vector resolution problems can be solved in a similar
manner. As a test of your understanding, utilize
trigonometric functions to determine the horizontal and
vertical components of the following initial velocity
values. When finished, click the button to check your
answers.

Practice A: A water
balloon is launched with a speed of 40 m/s at an angle of
60 degrees to the horizontal.

Practice B: A motorcycle
stunt person traveling 70 mi/hr jumps off a ramp at an
angle of 35 degrees to the horizontal.

Practice C: A springboard
diver jumps with a velocity of 10 m/s at an angle of 80
degrees to the horizontal.

As mentioned above, the point of resolving an initial
velocity vector into its two components is to use the values
of these two components to analyze a projectile's motion and
determine such parameters as the horizontal displacement,
the vertical displacement, the final vertical velocity, the
time to reach the peak of the trajectory, the time to fall
to the ground, etc. This process is demonstrated on the
remainder of this page. We will begin with the determination
of the time.

Determination of
the Time of Flight

The
time for a projectile to rise vertically to its peak (as
well as the time to fall from the peak) is dependent upon
vertical motion parameters. The process of rising vertically
to the peak of a trajectory is a vertical motion and is thus
dependent upon the initial vertical velocity and the
vertical acceleration (g = 9.8
m/s/s, down). The process of determining the time to
rise to the peak is an easy process - provided that you have
a solid grasp of the concept of acceleration. When first
introduced, it was said that acceleration
is the rate at which the velocity of an object changes.
An acceleration value indicates the amount of velocity
change in a given interval of time. To say that a projectile
has a vertical acceleration of -9.8 m/s/s is to say that
the vertical velocity changes
by 9.8 m/s (in the - or downward direction) each second.
For example, if a projectile is moving upwards with a
velocity of 39.2 m/s at 0 seconds, then its velocity will be
29.4 m/s after 1 second, 19.6 m/s after 2 seconds, 9.8 m/s
after 3 seconds, and 0 m/s after 4 seconds. For such a
projectile with an initial vertical velocity of 39.2 m/s, it
would take 4 seconds for it to reach the peak where its
vertical velocity is 0 m/s. With this notion in mind, it is
evident that the time for a projectile to rise to its peak
is a matter of dividing the vertical component of the
initial velocity (v_{iy}) by the acceleration of
gravity.

Once the time to rise to the peak of the
trajectory is known, the total time of flight can be
determined. For a projectile which lands at the same height
which it started, the total time of flight is twice the time
to rise to the peak. Recall from the last section of Lesson
2 that the trajectory of a
projectile is symmetrical about the peak. That is, if it
takes 4 seconds to rise to the peak, then it will take 4
seconds to fall from the peak; the total time of flight is 8
seconds. The time of flight of a projectile is twice the
time to rise to the peak.

Determination of
Horizontal Displacement

The horizontal displacement of a projectile is dependent
upon the horizontal component of the initial velocity. As
discussed in the previous part of this lesson, the
horizontal displacement of a projectile can be determined
using the equation

x = v_{ix} •
t

If a projectile has a time of flight of 8 seconds and a
horizontal velocity of 20 m/s, then the horizontal
displacement is 160 meters (20 m/s • 8 s). If a
projectile has a time of flight of 8 seconds and a
horizontal velocity of 34 m/s, then the projectile has a
horizontal displacement of 272 meters (34 m/s • 8 s).
The horizontal displacement is dependent upon the only
horizontal parameter which exists for projectiles - the
horizontal velocity
(v_{ix}).

Determination of
the Peak Height

A non-horizontally launched projectile with an initial
vertical velocity of 39.2 m/s will reach its peak in 4
seconds. The process of rising to the peak is a vertical
motion and is again dependent upon vertical motion
parameters (the initial vertical velocity and the vertical
acceleration). The height of the projectile at this peak
position can be determined using the equation

y = v_{iy} •
t + 0.5 • g • t^{2}

where v_{iy}
is the initial vertical velocity in m/s,
g is the acceleration of
gravity ( -9.8 m/s/s) and
t is the time in seconds
it takes to reach the peak. This equation can be
successfully used to determine the vertical displacement of
the projectile through the first half of its trajectory
(i.e., peak height) provided that the algebra is properly
performed and the proper values are substituted for the
given variables. Special attention should be given to the
facts that the t in the
equation is the time up to the peak and the g has a negative
value of -9.8 m/s/s.

Check
Your Understanding

Answer the following questions and click the button to
see the answers.

1. Aaron Agin is resolving velocity vectors into
horizontal and vertical components. For each case, evaluate
whether Aaron's diagrams are correct or incorrect. If
incorrect, explain the problem or make the correction.

2. Use trigonometric functions to resolve the following
velocity vectors into horizontal and vertical components.
Then utilize kinematic equations to calculate the other
motion parameters. Be careful with the equations; be guided
by the principle that "perpendicular components of motion
are independent of each other." PSYW

3. Utilize kinematic equations and projectile motion
concepts to fill in the blanks in the following tables.