Mathematics
Projectiles
Projectiles are objects that are launched into the air and follow a curved path due to the influence of gravity. In mathematics, the motion of projectiles can be modeled using equations that take into account the initial velocity, angle of launch, and gravitational acceleration. These equations can be used to predict the trajectory and landing point of a projectile.
Written by Perlego with AI-assistance
Related key terms
1 of 5
7 Key excerpts on "Projectiles"
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
• Find the time of flight and impact velocity of a projectile that lands at a different height from that of launch. • Calculate the trajectory of a projectile. Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration as a result of gravity. The applications of projectile motion in physics and engineering are numerous. Some examples include meteors as they enter Earth’s atmosphere, fireworks, and the motion of any ball in sports. Such objects are called Projectiles and their path is called a trajectory. The motion of falling objects as discussed in Motion Along a Straight Line 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, and our treatment neglects the effects of air resistance. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. We discussed this fact in Displacement and Velocity Vectors, where we saw that 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 resulting from gravity is vertical; thus, there is 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. It is not required that we use this choice of axes; it is simply convenient in the case of gravitational acceleration. In other cases we may choose a different set of axes. Figure 4.11 illustrates the notation for displacement, where we define s → to be the total displacement, and x → and y → are its component vectors along the horizontal and vertical axes, respectively.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v B AB , and the velocity of object B relative to object C is v B BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v B AB , the velocity of B relative to A is v B AB 5 2 v B AB . v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) v B AC 5 v B AB 1 v B BC (3) x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) FOCUS ON CONCEPTS Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via an online homework management program such as WileyPLUS or WebAssign.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically downward. If the trajectory of the projectile is near the earth’s surface, the ver- tical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x = 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is → v AB , and the velocity of object B relative to object C is → v BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is → v AB , the velocity of B relative to A is → v BA = − → v AB . → v AC = → v AB + → v BC (3) Focus on Concepts Additional questions are available for assignment in WileyPLUS. Section 3.3 Projectile Motion 1. The drawing shows projectile motion at three points along the trajectory. The speeds at the points are υ 1 , υ 2 , and υ 3 . Assume there is no air resistance and rank the speeds, largest to smallest. (Note that the symbol > means “greater than.”) (a) υ 1 > υ 3 > υ 2 (b) υ 1 > υ 2 > υ 3 (c) υ 2 > υ 3 > υ 1 (d) υ 2 > υ 1 > υ 3 (e) υ 3 > υ 2 > υ 1 2. Two balls are thrown from the top of a building, as in the draw- ing. Ball 1 is thrown straight down, and ball 2 is thrown with the same speed, but upward at an angle θ with respect to the horizontal. Consider the motion of the balls after they are released.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the ac- celeration due to gravity, which acts vertically downward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x = 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v → AB , and the velocity of object B relative to object C is v → BC . The velocity of A rel- ative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v → AB , the velocity of B relative to A is v → BA = −v → AB . v → AC = v → AB + v → BC (3) Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via WileyPLUS. Section 3.3 Projectile Motion 1. The drawing shows projectile motion at three points along the trajectory. The speeds at the points are υ 1 , υ 2 , and υ 3 . Assume there is no air resistance and rank the speeds, largest to smallest. (Note that the symbol > means “greater than.”) (a) υ 1 > υ 3 > υ 2 (b) υ 1 > υ 2 > υ 3 (c) υ 2 > υ 3 > υ 1 (d) υ 2 > υ 1 > υ 3 (e) υ 3 > υ 2 > υ 1 3. Two balls are thrown from the top of a building, as in the drawing. Ball 1 is thrown straight down, and ball 2 is thrown with the same speed, but upward at an angle θ with respect to the horizontal.
eBook - ePubSports Biomechanics
The Basics: Optimising Human Performance
- Prof. Anthony J. Blazevich(Author)
- 2017(Publication Date)
- Bloomsbury Sport(Publisher)
CHAPTER 3 PROJECTILE MOTIONBy the end of this chapter you should be able to:What is the optimum angle of trajectory or flight path (that is, the angle thrown relative to the ground) for a shot-putter aiming to throw the maximum distance? (Hint: not 45°.) What factors affect maximum throwing distance and to what degree?• List the factors that influence an object’s trajectory• Use the equations of projectile motion to calculate flight times, ranges and projection angles of Projectiles• Design a simple model to determine the influence of factors affecting projection range• Create a spreadsheet to speed up calculations to optimise athletic throwing performance• Complete a video analysis of a throw to optimise performanceProjectile motion refers to the motion of an object (for example a shot, ball or human body) projected at an angle into the air. Gravity and air resistance affect such objects, although in many cases air resistance is considered to be so small that it can be disregarded. A projected object can move at any angle between horizontal (0°) and vertical (90°) but gravity only acts on bodies moving with some vertical motion.Trajectory is influenced by the projection speed, the projection angle and the relative height of projection (that is, the vertical distance between the landing and release points; for example, in a baseball throw that lands on the ground, the vertical distance is the height above the ground from which the ball was released).FIG. 3.1 Tennis ball trajectory. Gravity accelerates the ball towards the ground at the same rate regardless of whether the tennis player leaves the ball to fall freely or hits it perfectly horizontally. However, the trajectory of the ball is different in these two circumstances.Projection speedThe distance a projectile covers, its range, is chiefly influenced by its projection speed. The faster the projection speed, the further the object will go. If an object is thrown through the air, the distance it travels before hitting the ground (its range) will be a function of horizontal velocity and flight time (that is, velocity × time, as you saw in Chapter 1 ). In Figure 3.1
eBook - PDFSports Math
An Introductory Course in the Mathematics of Sports Science and Sports Analytics
- Roland B. Minton(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 1 Projectile Motion Introduction Basketball star Stephen Curry launches a 3-point shot. As the ball traces its high arc toward the basket, fans rise to their feet in anticipation. Will it go in? Is it a little short? Similar tension accompanies a Jordan Spieth tee shot, an Andy Murray passing shot, a long football pass by Peyton Manning or Li-onel Messi, or a long fly ball by Mike Trout. We will analyze the flights of balls in this chapter as we explore the area of physics known as mechanics. Along the way, we will answer such ques-tions as: How does Blake Griffin hang in the air when dunking? What is the optimal angle to shoot a free throw? Why do golf balls have dimples? Does a knuckleball really dance? The answers are to be found in the funda-mentals of physics. Figuring with Newton Sir Isaac Newton (1643-1727) constructed a framework for the analysis of objects in motion. The second of his three Laws of Motion is the launching point for most of our investigations in this chapter. The shorthand version of Newton’s Second Law is F = ma where F is the sum of all forces acting on an object, m is the object’s mass, and a is the acceleration of the object. One of the most remarkable aspects of 1 2 Sports Math Newton’s Second Law is that it can also be written as F = m a , where F and a appear in bold to indicate that they are multidimensional vector quantities. We will return to this form of the equation when we look at motion in two and three dimensions. The mass m is a scalar (real number) that is related to weight: for earthbound sports, weight is approximately equal to mass times the gravitational constant g . To keep it simple, let’s start with one-dimensional motion; vertical motion, to be precise. In this case, the object’s position can be tracked by its height h above some reference point (e.g., the ground). We define velocity as the rate of change of position with respect to time.
eBook - PDFEngineering Mechanics
Dynamics
- L. G. Kraige, J. N. Bolton(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Therefore, every- thing covered in Art. 2 ∕ 2 on rectilinear motion can be applied separately to the x-motion and to the y-motion. Projectile Motion An important application of two-dimensional kinematic theory is the problem of projectile motion. For a first treatment of the subject, we neglect aerodynamic drag and the curvature and rotation of the earth, and we assume that the altitude change is small enough so that the acceleration due to gravity can be considered constant. With these assumptions, rectangular coordinates are useful for the tra- jectory analysis. For the axes shown in Fig. 2 ∕ 8, the acceleration components are a x = 0 a y = −g r = x i + y j y v = r ˙ = x ˙ i + y ˙ j y a = v ˙ = r ¨ = x ¨ i + y ¨ j y ¨ Path j i x i y j r A x y A v a v y v x a x a y FIGURE 2/7 Article 2/4 Rectangular Coordinates (x-y) 29 Integration of these accelerations follows the results obtained previously in Art. 2 ∕ 2a for constant acceleration and yields v x = ( v x ) 0 v y = ( v y ) 0 − gt x = x 0 + ( v x ) 0 t y = y 0 + ( v y ) 0 t − 1 2 gt 2 v y 2 = ( v y ) 0 2 − 2 g( y − y 0 ) In all these expressions, the subscript zero denotes initial conditions, frequently taken as those at launch where, for the case illustrated, x 0 = y 0 = 0. Note that the quantity g is taken to be positive throughout this text. We can see that the x- and y-motions are independent for the simple projectile conditions under consideration. Elimination of the time t between the x- and y- displacement equations shows the path to be parabolic (see Sample Problem 2 ∕ 6). If we were to introduce a drag force which depends on the speed squared (for exam- ple), then the x- and y-motions would be coupled (interdependent), and the trajec- tory would be nonparabolic.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
