![]() Figure 1 illustrates the notation for displacement, where\textbfabove the horizontal, as shown in Figure 4. (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. 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 fact was discussed in Chapter 3.1 Kinematics in Two Dimensions: An Introduction, where vertical and horizontal motions were seen to be independent. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. In this section, we consider two-dimensional projectile motion, such as that of a football or other object for which air resistance is negligible. The motion of falling objects, as covered in Chapter 2.6 Problem-Solving Basics for One-Dimensional Kinematics, is a simple one-dimensional type of projectile motion in which there is no horizontal movement. The object is called a projectile, and its path is called its trajectory. Projectile motion is the motionof an object thrown or projected into the air, subject to only the acceleration of gravity. ![]() Apply the principle of independence of motion to solve projectile motion problems.Determine the location and velocity of a projectile at different points in its trajectory.Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory.More sophisticated arguments for why the 45-degree launch angle yields the greatest range exist yet since they involve the use of calculus, they are not presented here.įor more information on physical descriptions of motion, visit The Physics Classroom Tutorial. The projectile launched at 45-degree does not win in either category, yet the fact that it is able to place a strong showing in each category contributes to its ability to achieve the greatest range. The projectile launched at 30-degrees has the greatest v x of all three launch angles yet its range is limited by the fact that the hang time is so short. As can be seen from the animation, the projectile launched at 60-degrees has the greatest hang time yet its range is limited by the fact that the v x is the smallest of all three angles. The range of a projectile is determined by two parameters - the initial value of the horizontal velocity component and the hang time of the projectile. The projectile launched at 30-degrees has the smallest v y, and as such the shortest hang time. The smaller the initial value of v y, the shorter the hang time. ![]() The "hang time" of a projectile is also determined by the initial value of the vertical velocity component. The projectile launched at 60-degrees has the greatest v y, and as such the greatest peak height. The greater the initial value of v y, the higher that a projectile will rise. ![]() The peak height of a projectile is determined by the initial value of the vertical velocity component. The cannonball launched at the 30-degree angle reached the ground first.Īn analysis of the velocity components for these three projectiles reveals reasons for these observations. The cannonball launched at a 60-degree angle had the highest peak height before falling. The cannonball launched at a 45-degree angle had the greatest range. Additionally, the velocity components (horizontal and vertical) are represented by arrows in the animation.Īs can be seen from the above animation, each cannonball follows a parabolic path. How will the trajectories of the three cannonballs compare? Which cannonball will have the greatest range? Which cannonball will reach the highest peak height before falling? Which cannonball will reach the ground first? The animation below depicts such a situation. Imagine as well that the cannonballs do not encounter a significant amount of air resistance. The launch speed is held constant only the angle is changed. Imagine a cannonball launched from a cannon at three different launch angles - 30-degrees, 45-degrees, and 60-degrees. Multimedia Studios » Vectors and Projectiles » Maximum Range
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