Describing Projectiles With Numbers:
(Horizontal and Vertical Velocity)
So far in Lesson 2 you have learned the
following conceptual notions about projectiles.
· A
projectile is any object upon which the only force is gravity,
· Projectiles
travel with a parabolic trajectory due to the influence of gravity,
· There are
no horizontal forces acting upon projectiles and thus no horizontal
acceleration,
· The
horizontal velocity of a projectile is constant (a never changing in value),
· There is a
vertical acceleration caused by gravity; its value is 9.8 m/s/s, down,
· The
vertical velocity of a projectile changes by 9.8 m/s each second,
· The
horizontal motion of a projectile is independent of its vertical motion.
In this portion of Lesson 2 you will learn how to describe
the motion of projectiles numerically. You will learn how the numerical values
of the x- and y-components of the velocity and displacement change
with time (or remain constant). As you proceed through this part of Lesson 2,
pay careful attention to how a conceptual understanding of projectiles
translates into a numerical understanding.
Consider again the cannonball launched by a
cannon from the top of a very high cliff. Suppose that the cannonball is
launched horizontally with no upward angle whatsoever and with an initial speed
of 20 m/s. If there were no gravity, the cannonball would continue in motion at
20 m/s in the horizontal direction. Yet in actuality, gravity causes the
cannonball to accelerate downwards at a rate of 9.8 m/s/s. This means that the
vertical velocity is changing by 9.8 m/s every second. If a vector diagram (showing
the velocity of the cannonball at 1-second intervals of time) is used to
represent how the x- and y-components of the velocity of the cannonball is
changing with time, then x- and y- velocity vectors could be drawn and their magnitudes labeled. The lengths of the vector arrows are
representative of the magnitudes of that quantity. Such a diagram is shown
below.
The important concept depicted in the above vector diagram is
that the horizontal velocity remains constant during the course of the
trajectory and the vertical velocity changes by 9.8 m/s every second. These
same two concepts could be depicted by a table illustrating how the x- and
y-component of the velocity vary with time.
Time |
Horizontal Velocity |
Vertical Velocity |
0 s |
20 m/s, right |
0 |
1 s |
20 m/s, right |
9.8 m/s, down |
2 s |
20 m/s, right |
19.6 m/s, down |
3 s |
20 m/s, right |
29.4 m/s, down |
4 s |
20 m/s, right |
39.2 m/s, down |
5 s |
20 m/s, right |
49.0 m/s, down |
The numerical information in both the diagram
and the table above illustrate identical points - a projectile has a vertical
acceleration of 9.8 m/s/s, downward and no horizontal acceleration. This is to
say that the vertical velocity changes by 9.8 m/s each second and the
horizontal velocity never changes. This is indeed consistent with the fact that there is a vertical
force acting upon a projectile but no horizontal force. A vertical force causes a vertical acceleration - in this case, an
acceleration of 9.8 m/s/s.
But what if the projectile is launched upward at an angle to
the horizontal? How would the horizontal and vertical velocity values change
with time? How would the numerical values differ from the previously shown
diagram for a horizontally launched projectile? The diagram below reveals the
answers to these questions. The diagram depicts an object launched upward with
a velocity of 75.7 m/s at an angle of 15 degrees above the horizontal. For such
an initial velocity, the object would initially be moving 19.6 m/s, upward and
73.1 m/s, rightward. These values are x- and y-components of the
initial velocity and will be discussed in more detail in the next part of
this lesson.
Again, the important concept depicted in the above diagram is
that the horizontal velocity remains constant during the course of the
trajectory and the vertical velocity changes by 9.8 m/s every second. These
same two concepts could be depicted by a table illustrating how the x- and
y-component of the velocity vary with time.
|
Horizontal Velocity |
Vertical Velocity |
0 s |
73.1 m/s, right |
19.6 m/s, up |
1 s |
73.1 m/s, right |
9.8 m/s, up |
2 s |
73.1 m/s, right |
0 m/s |
3 s |
73.1 m/s, right |
9.8 m/s, down |
4 s |
73.1 m/s, right |
19.6 m/s, down |
5 s |
73.1 m/s, right |
29.4 m/s, down |
6 s |
73.1 m/s, right |
39.2 m/s, down |
7 s |
73.1 m/s, right |
49.0 m/s, down |
The numerical information in both the diagram
and the table above further illustrate the two key principles of projectile
motion - there is a horizontal velocity that is constant and a vertical
velocity that changes by 9.8 m/s each second. As the projectile rises towards
its peak, it is slowing down (19.6 m/s to 9.8 m/s to 0 m/s); and as it falls
from its peak, it is speeding up (0 m/s to 9.8 m/s to 19.6 m/s to ...).
Finally, the symmetrical nature
of the projectile's motion can be seen in the diagram above: the vertical speed one second
before reaching its peak is the same as the vertical speed one second
after falling from its peak. The vertical speed two seconds before reaching its peak is the same as the vertical speed two
seconds after falling from its peak. For non-horizontally launched projectiles,
the direction of the velocity vector is sometimes considered + on the way up
and - on the way down; yet the magnitude of the vertical velocity (i.e.,
vertical speed) is the same an equal interval of time on
either side of its peak. At the peak itself, the vertical velocity is 0 m/s;
the velocity vector is entirely horizontal at this point in the trajectory.
These concepts are further illustrated by the diagram below for a
non-horizontally launched projectile that lands at the same height as which it
is launched.
The above diagrams, tables, and discussion pertain to how the
horizontal and vertical components of the velocity vector change with time
during the course of projectile's trajectory. Another vector quantity that can
be discussed is the displacement. The numerical description of the displacement
of a projectile is discussed in the next section of Lesson 2.