PHYS 251 - Introduction to Computer Techniques in Physics

Physics Application: Simple Projectile Motion

Notation: Equation numbers are placed in parentheses, (Eq. 1), at the end of the sentence preceding the mathematical presentation of the equation. They should not be read as if they are part of the sentence. When reference is made to an equation, the word "equation" will be spelled out and not abbreviated. Mathematical and Greek symbols are not in all cases precise, since they may depend on the browser used.

Basic Equations for Simple Projectile Motion

For the simple problem of the projectile starting and ending at the same vertical height, let x be horizontal position and y the vertical position with both measured from the launch position and x positive to the right and y positive upward. The projectile's path is a parabola confined to a plane so that projectile motion is an example of two-dimensional motion. Projectile motion is also an example of motion with constant acceleration in the vertical direction as provided by gravity. In addition, we will assume no acceleration in the x direction. Thus in mathematical form we have for the acceleration (Eq. 1)

$$a = 0,

or in component form where acceleration is the time-rate of change of velocity

$a$_x = d v_x / d t = 0,
a_y = d v_y / d t = - g,

where g = acceleration due to gravity acting in the vertical direction. These equations can immediately be integrated with respect to time in both the x and y components to obtain the velocity components (Eq. 2) as

$v$_x = v_0x = v₀ cos (q ),
v_y = v_0y - g t = v₀ sin (q )- g t,

where v₀ = launch velocity, q = launch angle relative to the horizontal, v_0x = a constant of integration and the initial velocity component in the x direction, and v_0y = a constant of integration and the initial velocity component in the y direction. A second integration with respect to time in both the x and y components yields the position components (Eq. 3)

$x\; =\; x$₀ + v_0x t = x₀ + v₀ cos (q ) t,
y = y₀ + v_0y t - ¹/₂ g t² = y₀ + v₀ sin ( q ) t - ¹/₂ g t²,

where x₀ = constant of integration or the initial x position = 0, and y₀ = constant of integration or the initial y position = 0.

Time of Flight, Range and Maximum Height

There are three quantities important in analyzing simple projectile motion. They are:

1. the time of flight, T, which is the time from launch ( y = 0 ) to impact ( y = 0, again ),
2. the range, R, which is the maximum horizontal distance traversed by the projectile,
3. and the maximum height, H, which is the height of the peak of the parabolic path and half of the horizontal range.

In mathematical form, these quantities can be calculated by the following equations (Eq. 4)

$T\; =\; 2\; v$_0y / g = 2 v₀ sin (q ) / g,
R = ( v₀² / g ) sin ( 2 q ),
H = v_0y² / 2 g .

Projectile Motion Solving ODEs

An additional discussion of the physics of projectile motion is given at the following link (Physics Application: Projectile Motion). This discussion involves the solution of coupled ordinary differential equations that characterize projectile motion using Euler's Method of solution.