Projectile Motion

10 Key excerpts on "Projectile Motion"

• 

  eBook - PDF

  University Physics Volume 1

  • 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.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Sports Biomechanics

  The Basics: Optimising Human Performance

  • Prof. Anthony J. Blazevich(Author)
  • 2017(Publication Date)
  • Bloomsbury Sport

    (Publisher)

  CHAPTER 3 Projectile Motion

  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?

  By the end of this chapter you should be able to:

  • 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 performance

  Projectile 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 speed

  The 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

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Workshop Physics Activity Guide Module 1

  Mechanics I

  • Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  It’s worth noting that the gravitational force, which acts vertically, has no effect on our horizontal motion experiments like the cart on a track. 12 In other words, motions in the horizontal and vertical directions are independent of one another. We will observe this direction independence shortly, when we inves- tigate the motion of an object that is launched near the surface of Earth. Such motion is commonly referred to as Projectile Motion. Before studying motion in two dimensions, we would like to do an experiment that demonstrates the physical nature of a constant, continuous force in one dimension. For example, a dropped ball experiences a constant downward force due to gravity. We can create a horizontal analog of this motion by continuously tapping a ball in one direction on a flat surface. The similarity between a falling ball and a tapped ball will help us study Projectile Motion, in which an object falls vertically while simultaneously moving horizontally. For the measurements described below, you will use a twirling baton with a rubber tip to tap a bowling ball and watch its motion. You should have the fol- lowing equipment available: • 1 bowling ball (or other heavy ball) • 1 twirling baton (or other stick used to tap the ball) 11 Clearly, Earth cannot be seen as flat from their perspective. 12 We did need to be careful to keep the motion only in the horizontal direction—any “tilt” to the cart track would have led to an acceleration “downhill” caused by the gravitational force. UNIT 6: GRAVITY AND Projectile Motion 183 Find a stretch of smooth, level floor over which the ball can roll for some dis- tance (a hallway can be used if the classroom is not large enough).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Sports 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Engineering 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The motion can be analysed by treating the x and y components of the four vectors separately and realising that the time t is the same for each component. When the acceleration is constant, the x components of the dis- placement, the acceleration, and the initial and final velocities are related by the equations of kinematics, and so are the y components. x Component y Component v x = v 0x + a x t (3.3a) v y = v 0y + a y t (3.3b) x = 1 2 ( v 0x + v x ) t (3.4a) y = 1 2 ( v 0y + v y ) t (3.4b) x = v 0x t + 1 2 a x t 2 (3.5a) y = v 0y t + 1 2 a y t 2 (3.5b) v x 2 = v 0 2 x + 2a x x (3.6a) v y 2 = v 0 2 y + 2a y y (3.6b) The directions of the components of the displacement, the accel- eration, and the initial and final velocities are conveyed by assigning a plus (+) or minus (−) sign to each one. 3.3 Analyse Projectile Motion to predict future or past values of variables. Projectile Motion is an idealised 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 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 Apply relative velocity equations. 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).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Guide to Mechanics

  • Philip Dyke, Roger Whitworth(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  If the velocity of projection is V at an angle to Ox , as illustrated in Figure 5.12, then the velocity of projection can be written as the vector: u ˆ V cos i ‡ V sin j The horizontal motion is unaffected by the acceleration and the component of velocity in that direction is constant. The upwards vertical motion is subject to an acceleration of g . The acceleration vector is written in the form: a ˆ g j For a given point r relative to O at time t on the trajectory, we can write: r ˆ u t ‡ 1 2 a t 2 The vector diagram in Figure 5.13 shows the dependence of the position vector r on the vectors u and a . This gives, for a given point on the path with coordinates ( x , y ): x i ‡ y j ˆ V cos † t i ‡ V sin † t j † ‡ 1 2 g j † t 2 O y V x Figure 5.12 The trajectory 124 Guide to Mechanics O r u t ½ a t 2 Figure 5.13 A triangle of vectors illustrating the equation r ˆ u t ‡ 1 2 a t 2 The set of equations: x ˆ V cos † t and y ˆ V sin † t 1 2 gt 2 represents the parametric equations of the trajectory. Making t the subject of the first of these formulae, we obtain: t ˆ x V cos This enables us to derive: y ˆ x tan gx 2 sec 2 2 V 2 5 : 26 † as the Cartesian equation for the trajectory. From this equation, we can see that, for a given V and , the trajectory is a parabola. Having once determined this we can use any of the properties of a parabola to discuss Projectile Motion. The most important of these is the symmetry of the curve about a vertical line through its maximum value, which is of course its greatest height. Using this symmetry property, we can make the following observations about Projectile Motion: (a) The greatest height is the maximum value of y . (b) The range is the value of x for which y ˆ 0. (c) The range is twice the x value to the greatest height. (The time to the range is twice the time to the greatest height, which follows directly.) (d) All heights are symmetrical about the greatest height's horizontal position.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  3.2 Equations of Kinematics in Two Dimensions Motion in two dimensions can be described in terms of the time t and the x and y components of four vectors: the displacement, the acceleration, and the initial and final velocities. The x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist. The motion can be analyzed by treating the x and y components of the four vectors separately and realizing that the time t is the same for each component. 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) Concept Summary 65 66 Chapter 3 | Kinematics in Two Dimensions When the acceleration is constant, the x components of the displacement, the acceleration, and the initial and final velocities are related by the equations of kinematics, and so are the y components: 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) The directions of the components of the displacement, the acceleration, and the initial and final velocities are conveyed by assigning a plus (1) or minus (2) sign to each one. 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Applied Mathematics

  Made Simple

  • Patrick Murphy(Author)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  With this in mind we decide to consider the displacement, velocity, and acceler-ation of the body in two parts: one vertical and the other horizontal. (3) Equations of Motion Since the acceleration is constant we may use the equations of motion that we deduced in Chapter Four. We recall the following notation and results: Initial velocity is u Velocity at time t is ν Acceleration is a Displacement is s ν = u + at; v 2 = u 2 -f 2as; s = ut + at 1 Projectiles 167 These equations were discussed in terms of motion in a straight line, and we know that the path of a projectile is not a straight line. However, there is no difficulty here as we shall be applying the equations to the horizontal and vertical components of the motion separately. We start by considering a numerical case, that of a body projected freely under gravity only, with an initial velocity of 100 m s 1 , at an angle of 60° to the vertical. Its path is represented by Fig. 146. The correct name for such a curve is a parabola and the path described by the projectile is called its trajectory. First of all we identify the notation of the question: Horizontally Vertically Displacement: x y Initial velocity (w) : 100 cos 60° 100 sin 60° Velocity at time (v): v x v y Acceleration (a): 0 — g f I 0 0 m s -y P (x,y) v y Ί > 0 A 60 ° f L Β A X Fig. 146 Thus after time t the body is at P, whose position is given by (x, y) and, the components of the velocity are v x and v y . Notice that v y will be negative since the body is descending at P. The result of substituting the details above into the equations of motion are as follows: Motion Horizontally v x = 100 cos 60° = 50 χ = (100 cos 60°)r = 50 ί Motion Vertically Vy = (100sin60°) -gt (^ = 9 8 1 or 9-80 m s 2 as discussed in V = (100 sin 60°) 2 —2gy Chapter Four) y = ( 1 0 0 s i n 6 0 ° ) i -i ^ 2 These five equations will tell us all we need to know about the motion. All we have to do is choose the right equations for the information we require.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Physics, Volume 1

  • Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  As an example, imagine you are swinging a ball on a string in a horizontal plane, as in Fig. 4-14. (We neglect the drag force and the force of gravity for the time being.) As you swing the ball, your fingers are exerting a force on the string (and the string in turn exerts a force on the ball). If you were to loosen your grip on the string slightly, the string would slide between your fingers and the ball would move away from the center of the circle, so to prevent this D B ,   0  60.  0  60, 4-5 Uniform Circular Motion 73 x D v 0 m g v 0 y Figure 4-12. A projectile in motion. It is launched with ve- locity v 0 at an angle  0 with the horizontal. At a certain time later its velocity is at the angle . The weight and the drag force (which always points in a direction opposite to are shown at that time. v B ) v B * You can find more information about this calculation in “Trajectory of a Fly Ball,” by Peter J. Brancazio, The Physics Teacher, January 1985, p. 20. For an interesting collection of articles about similar problems, see The Physics of Sports, edited by Angelo Armenti, Jr. (American Institute of Physics, 1992). See http://www.physics.uoguelph.ca/fun/JAVA/trajplot/ trajplot.html for an interesting program that allows you to display the trajec- tories of a projectile for various choices of launch angle and air resistance. Figure 4-13. Projectile Motion with and without a drag force, calculated for 45 m/s and  0  60°. v 0  y (m) t = 5 s t = 5 s t = 6 s t = 6 s t = 7 s t = 4 s t = 4 s t = 3 s t = 3 s t = 2 s t = 2 s t = 1 s t = 1 s 50 x (m) 0 = 60 ° 100 With drag force Without drag force Range 179 m Range 72 m – 60 ° – 79 ° Figure 4-14. A ball on a string is whirled in a horizontal cir- cle. Vectors representing the velocity and the force of the string on the ball are shown at three different instants. v v F BS F BS F BS v from happening your fingers must be exerting an inward force on the string.

  Sign up to read

  Learn more about book