One of the most important applications of the formulations for studying
motion in two dimensions is the study of projectile motion. This motion occurs
when an object, near the surface of the Earth, is through in any direction (not
necessarily straight up) and the air
resistance is neglected. For the motion of an object under the previous
conditions, the vertical component of the motion obey
the formulations used for free fall motion while the horizontal component of the motion
corresponds to motion with constant velocity.
Quantity 
Horizontal
Component 
Vertical Component 
Acceleration 
There is not acceleration in this direction,
. 
The acceleration is the acceleration of gravity,

Velocity 
The velocity of the object in the horizontal
direction is with constant velocity,
. 
Since the vertical motion is motion with constant
acceleration, the same formulas used for free fall motion can be applied
to this motion. 
Velocity (average) 

Velocity (average) 

Velocity (instantaneous) 

Time Independent Equation 

Displacement 
There is only one equation that relates the
displacement, velocity, and time for the component of the motion in the horizontal
direction, 
The displacement along the vertical component of the
motion is given, again, by the same formula as that for free fall 


Displacement 

As a consequence of the equations for the horizontal
component of the motion and the vertical component of the motion, the trajectory
described by the projectile correspond to part of a parabola independent of the
initial conditions. In fact, the horizontal and vertical components of the
displacement are written in the parametric form of a parabola where the
parameter is the time. An explicit parabola can be obtained by solving for the
time in the formula for the horizontal displacement,
,
and substituting this solution in the formula for the vertical displacement,
.
In the last expression, the velocity in the horizontal direction is constant.
The previous relation is a quadratic function of the vertical position as a
function of the horizontal position.

Vector Relations

In addition to the previous group of formulas, relations between different
vectorial quantities and their correspondent components can be found,
Initial Velocity 
Velocity at Time
t 


In the above figure,
represents the vector initial velocity. This vector is defined as
where q_{0} is the angle with
respect to the horizontal plane. This vector can be broken down into two
vector components one along the horizontal direction and the second one
along the vertical direction. These two vectors are defined as
and
Notice that the previous signs are also obtained from
the corresponding trigonometric expressions,
for the horizontal component and
for the vertical component. Remember that the values of the function
cosine are negative for arguments greater than 90^{0} and
smaller than 180^{0} while the values of the function sine are
negative for arguments greater than 180^{0} and smaller than 360^{0}.
Knowing the components of the vector initial
velocity, the magnitude and direction of the vector initial velocity can
be obtained from the following two relations
In order to calculate the angle
q_{0} is very important to track
the sign of the components of the initial velocity vector.

The vector velocity is tangent to the parabola
representing the projectile motion of the object at any instant of time.
By definition, the
instantaneous velocity is the tangent to the curve representing the
path followed by the object. The components of the velocity are
calculated using the formulas
On the other side, knowing the components, the
magnitude and direction of the velocity can be calculated using

Displacement at Time
t 

The displacement of the projectile is represented by
the vector
.
The magnitude and direction of this vector are given by
At the same time, the components of the displacement
along the horizontal and vertical direction can be calculated from

In the previous table, the most useful relations for solving
problems are those associated with the initial velocity of the projectile.

Maximum Height and Range

In the study of projectile motion, there are two additional
quantities that are important to identified, the maximum height and the range of
a projectile.
Maximum Height 
Range 


The maximum height is obtained at the point where the
vertical component of the velocity vanishes. At this point, the velocity
of the projectile is identical in magnitude and direction to the
horizontal component of the velocity. From the time independent formula
above (Time Independent Equation), the maximum height can be calculated,

The range of the projectile is obtained when the
object returns to the ground. In most cases (but not always), the ground
correspond to y = 0 point where the projectile will impact the
ground. However, if the terrain is not flat, the returning point may be
different than the launching point of the projectile. Thus, in most
cases, the range is calculated using the point y = 0 as a
reference for the returning point. In the formula for the horizontal
displacement the only unknown is the time taken by the projectile to
return to the ground. This time is calculated by using the vertical
component of the displacement at the value of y representing the
ground level,

The previous formula becomes an equation for the
time,
This quadratic equation can be solved by simple
factorization of the time.
There are two solutions for the previous equation,
t = 0 and
.
The first solutions correspond to the time when the projectile was
launched, t = 0. The second solution corresponds to the desired
solution. Thus, substituting this time into the equation for the
horizontal displacement, the range of the projectile can be calculated.
This range can be written in terms of the magnitude
and direction of the initial velocity,
Range of the Projectile
The last relation can be simplified farther using the
trigonometric relation,
.
The following expression is the standard form of the range for a
projectile,
Standard form of the Range 


For a given magnitude for the initial velocity, there is an
angle for which the range is actually maximum. This angle corresponds to the
maximum of the trigonometric function sine which is a multiplication factor of
the Standard form of the Range (see above). The maximum of the function sine can be obtained
graphically from the figure below,

Graph of the trigonometric function sine for one
period. The maximum value of the function is obtained when the angular
argument is 90^{0}. At this point, the value of the
trigonometric function sine is
sin 90^{0}
= 1.
Therefore, in the case of the projectile motion, the maximum range
of the projectile is obtained when


The plotting of the trigonometric factor associated
with the formula labeled Range
of the Projectile is presented at the graph on the left. Notice that
the period of the factor sin q
cos q is
180^{0} rather than 360^{0} as it is the case for the
individual trigonometric functions plotted in the figure. Thus, between 0^{0} and
90^{0}, the values of the sin q
cos q factor
for different angles are repeated. In fact,
cos q = sin (90^{0}
 q) leading to sin q
cos q = sin
q sin (90^{0}
 q). In the last
expression, the value of the trigonometric factor are the same for
a = q as well as
for b = (90^{0} 
q). Therefore, adding the two angles, it is
obtained
a +
b = q +(90^{0}
 q)
a +
b = 90^{0}
In conclusion, the range of the projectile is the
same for two launching angles that are the complement of each other. 
The graphs below show the trajectory of the projectile for different
launching angles. The value for the magnitude of the initial velocity is
the same for all launching angles. Notice that the ranges for the projectiles
corresponding to
complementary angles are the same. 

