Download A Little Background on Projectile Motion

Gummy Bears Using Projectile Motion
Created by: Jennifer Fillingim
Grade Level: 8th or 9th grade
Objective: The aim of this handout is to provide a conceptual understanding
of projectile motion. Once students grasp the equations that describe both
the vertical and horizontal displacement of an object, they will use those
concepts to project gummy bears through the air in a manner that maximizes
the horizontal distance that the bear travels.
Background Information:
Because an object in projectile motion is moving in both vertical and
horizontal directions, we must analyze its motion based on the motion’s
vertical and horizontal components.
BASIC KINEMATICS FORMULAS:
v = vo + at
(velocity) = (initial velocity) + (acceleration)(time)
s = vot + (1/2)at2
distance = (initial velocity)(time) + (1/2)(acceleration)(time)2
Looking at velocity components in projectile motion:
sin  = (vy/v)  vy = v (sin )
cos  = (vx/v)  vx = v (cos )
SO, the initial velocity of the object being fired can be analyzed in terms of
its initial horizontal velocity and its initial vertical velocity based on the sine
and cosine of the particular angle .
What’s the catch?
The “catch” is that while gravity is acting upon the vertical displacement of
the object to pull it down, there is no outside force acting on the object’s
horizontal displacement (if we are neglecting air resistance and wind and
such, which we typically do in such problems). In fact, it was Galileo who
discovered that because of the object’s property of inertia, the horizontal
motion is uniform and constant. (An object’s property of inertia essentially
says that an object in motion tends to stay in motion unless acted upon by an
outside force; or an object at rest tends to stay at rest unless acted upon by
an outside force)
So, what does that mean?
Let’s consider our kinematics equations again:
s = vot + (1/2)at2
To solve for vertical displacement, we can substitute using our trig equation
vy = v (sin ) :
y = [vo(sin )] t + (1/2)(-9.8 m/s2)(t2)
where (-9.8 m/s2) is the measured force of gravity acting against the
motion of the object and vo(sin ) represents the vertical component
of initial velocity.
x = [ vo(cos )] t + (1/2)(0)(t2) = [vo (cos )] t
because there is no acceleration or deceleration in the horizontal
direction! Gravity only affects the object’s vertical displacement!
Therefore, the displacement of the object is completely dependent on
(1)
(2)
the initial velocity (vo)
the angle at which the object was fired
Thus, in order to maximize the displacement of the object, we either have to
(1)
(2)
increase the initial velocity
OR
keep the initial velocity constant and change the angle of firing
Comments:
The way I understand the intention of the “Projectile Gummy Bears” project,
the students will maximize the displacement of the object through the second
option. That is, they will keep the initial velocity constant while changing
the angle of projection. Ideally, each student is able to maintain a constant
initial velocity. In reality, however, the velocity is altered through changes
in the angle of its vertical and horizontal components.
As an interesting side note, it turns out that the ideal angle for a projectile
motion experiment such as this is 45 due to the interaction of the sine and
cosine functions.