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Description

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.

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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 q0 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 900 and smaller than 1800 while the values of the function sine are negative for arguments greater than 1800 and smaller than 3600.

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 q0 is very important to track the sign of the components of the initial velocity vector.

v0y v0x + -
+
-

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.

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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

 

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Maximum Range and Duality 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 900. At this point, the value of the trigonometric function sine is
sin 900 = 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 1800 rather than 3600 as it is the case for the individual trigonometric functions plotted in the figure. Thus, between 00 and 900, the values of the sin q cos q  factor for different angles are repeated. In fact, cos q = sin (900 - q) leading to sin q cos q = sin q sin (900 - q). In the last expression, the value of the trigonometric factor are the same for  a = q as well as for b = (900 - q). Therefore, adding the two angles, it is obtained

a + b = q +(900 - q)

a + b = 900

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.
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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com