28 4.4 Projectile Motion
Summary
- Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory.
- Determine the location and velocity of a projectile at different points in its trajectory.
- Apply the principle of independence of motion to solve projectile motion problems.
Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. Since the object or body is under the effects of a constant acceleration (-9.8m/s2 in the vertical and 0 in the horizontal plane) its trajectory is predictable based on the magnitude and direction of its initial velocity at take-off. The object or body is called a projectile, and its path is called its trajectory. The motion of falling objects is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this section, we consider two-dimensional projectile motion, such as that of a football or other object for which air resistance is negligible.
PHET EXPLORATIONS: PROJECTILE MOTION
Blast a Buick out of a cannon! Learn about projectile motion by firing various objects. Set the angle, initial speed, and mass. Add air resistance. Make a game out of this simulation by trying to hit a target.
The trajectory of a projectile takes on a parabolic shape. The very top of the trajectory is called the apex. If a projectile takes off and lands at the same height, the trajectory is symmetrical. This means that the projectile travels the same distance in both the vertical and horizontal plane on the way up, as on the way down. The time for the projectile to reach the apex, is the same as the time for the projectile to come back to the initial height.
The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. Vertical and horizontal motions are independent. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible, because acceleration due to gravity is vertical—thus, there will be no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the x-axis and the vertical axis the y-axis. Figure 1 illustrates the notation for displacement, where [latex]\vec{\textbf{d}}[/latex] is defined to be the total displacement and x and y are its components along the horizontal and vertical axes, respectively. The magnitudes of these vectors are x, and y. (Note that in the last section we used the notation [latex]\vec{\textbf{A}}[/latex] to represent a vector with components Ax and Ay. If we continued this format, we would call displacement [latex]\vec{\textbf{d}}[/latex] with components dx and dy.
Of course, to describe motion we must deal with velocity and acceleration, as well as with displacement. We must find