Acceleration in Projectile Motion

6 Key excerpts on "Acceleration in Projectile Motion"

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  y as functions of time directly and separately, and calculate the trajectory, the mathematical description of the path of the projectile that gives the projectile’s height as a function of horizontal position. We will also calculate how far the projectile travels horizontally before landing, how much time it spends in the air, and its maximum height.

  No Air Resistance

  In the absence of air resistance, the only influence on the projectile is gravity. Near the Earth’s surface, the acceleration due to gravity is constant so we can use Eqs. (2.5) and (2.6) to describe the projectile’s motion. Since we will describe both vertical and horizontal motions, we need to alter those equations to distinguish the two motions.

  Here are the general horizontal equations of motion.

  (4.1a)

  (4.1b)

  (4.1c)

  The subscript “x” on the velocity and acceleration indicates the horizontal components of those vectors.

  A similar alteration produces the vertical equations of motion.

  (4.2a)

  (4.2b)

  (4.2c)

  These six equations describe all 2-dimensional motion with constant acceleration. Gravity always pulls objects down, never sideways, never up. The acceleration due to gravity is constant, with a downward vertical component only

  (4.3a)

  (4.3b)

  where g = 9.806 7 m/s2 . This condition is valid for motion in a vacuum or at speeds small enough that the effects of air resistance are negligible. Since the acceleration is constant, we can apply Eqs. (4.1a) and (4.2a) to the position of a projectile.

  (4.4a)

  (4.4b)

  You can use these equations to calculate the projectile’s position (its x and y coordinates) or its displacement (Δx = x - x0 and Δy = y - y0 ) as functions of time.

  Example 4.1

  And the pitch…

  How much time does a 95 mile-per-hour fastball (thrown horizontally) take to get to home plate? How far does the ball drop? Ignore air resistance and spin effects.

  Solution.

  The ball’s horizontal speed is constant vox = 95 mi/h = 42.469 m/s and its horizontal displacement is Δx = 60.5 ft = 18.440 m. Using Eq. (4.4a) , the time it takes the ball to get to home plate is t = Δx/vox . Create and Evaluate

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  BIOS Instant Notes in Sport and Exercise Biomechanics

  • Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
  • 2007(Publication Date)
  • Routledge

    (Publisher)

  However, this is a clear illustration of why it is easier for astronauts to jump higher while they are on the surface of the Moon (i.e., the reason why you see them able to take large leaps and bounds while Moon walking). However, for the purpose of studying biomechanics the value for the Earth’s gravitational acceleration should be considered as 9.81 m/s 2. Vertical projection Gravity, as we have seen previously, is an external force that affects only the vertical component of projectile motion. In previous sections within this text we have seen that gravity does not affect the horizontal component of projectile motion. The effect of the force of gravity in the balance of the net forces acting is often expressed as an acceleration value (9.81 m/s 2) and in the understanding of vertical projection it is important to represent velocities and accelerations with directional components (as they are vector quantities that have both magnitude and direction). If we throw a ball into the air, and we were able to throw this ball perfectly vertically upwards (although in practice this is not so easy to achieve) gravity would be acting on the ball (actually gravity is acting on both us and the ball all the time). The acceleration due to gravity in this case would be expressed as –9.81 m/s 2. The minus sign would denote that gravity is acting vertically downward (i.e., trying to pull the ball downward towards the Earth’s mass center or trying to slow down its vertical ascent when we throw it into the air). Fig. B5.2 helps to illustrate this exercise in more detail. In Fig. B5.2 the ball leaves our hand with a specific amount of upward vertical (+ve) velocity. This is created from how much net force was eventually applied to the ball and for how long it was applied (i.e., net vertical impulse = force × time = change in momentum (vertical momentum))

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  eBook - ePub

  Describing Motion

  The Physical World

  • Robert Lambourne(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Figure 2.20 .

  Figure 2.20 Some examples of projectiles being launched.

  In what follows we shall ignore air resistance and model all projectiles as particles; even so, many of the results we obtain will be approximately correct and will certainly be of considerable interest.

  The study of projectiles has long been considered an important problem in science. In medieval times, the paths of cannon-balls and arrows were of interest to the military, so there were very practical reasons for seeking to understand the way in which projectiles moved. The first person to achieve such an understanding was the Italian scientist Galileo Galilei.

  The essential point which others had missed but which Galileo grasped was this: The horizontal and the vertical movements of a projectile are independent, apart from the fact that they must both have the same duration. In particular in the absence of air friction:

  1. The vertical component of a projectile’s motion is an example of uniformly accelerated motion in which the constant acceleration caused by gravity is directed downward and has magnitude g (with the approximate value 9.8 m s−2 ).
  2. The horizontal component of a projectile’s motion is an example of uniform motion in which the constant horizontal velocity is determined when the projectile is launched.

  So a projectile travels with constant velocity in the horizontal direction, during the time that it takes to rise and fall under constant downward acceleration in the vertical direction.

  By combining this principle with information about the initial position and velocity, Galileo was able to predict the motion of projectiles such as cannon-balls. Galileo’s many scientific achievements, of which this may be the greatest, have caused a number of commentators to regard him as the first true physicist in the modern sense.

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  Sports Biomechanics

  The Basics: Optimising Human Performance

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

    (Publisher)

  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 , you can see that a ball thrown in the air by a tennis player will hit the ground at the same time regardless of whether it is hit horizontally by the player or allowed to fall freely but the trajectory of the ball is different.

  If the projectile moves only vertically (for example, a ball thrown straight upwards), its projection speed will determine the height it reaches before gravity accelerates it back towards the Earth. If we don’t take air resistance into account, gravity accelerates all objects at the same rate: 9.81 m·s-2 barring some regional variations around the planet* . This is about the same acceleration a lion can achieve or twice the acceleration of the fastest humans. To get an idea of how fast it is, drop a small ball from a height of a few metres and watch it accelerate as it falls.

  What might position (displacement), velocity and acceleration graphs look like for a ball thrown vertically? Projection angle

  The angle of projection is also an important factor affecting projectile range. If an object is projected vertically, it will land back at its starting point, after gravity has pulled it back to Earth (remember, the object is accelerated positively the whole way if ‘down’ is assigned the positive direction). So, its range is zero. If the object is projected horizontally from ground level, it will not get airborne, so again its range is zero. It can also be projected at angles between 0° and 90°, where it will travel both vertically and horizontally. At a projection angle of 45° the object will have an equal magnitude of vertical and horizontal velocity and its range will be maximised, as you can see in Figure 3.2

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  eBook - ePub

  Mathematics and the Physical World

  • Morris Kline(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  x in the horizontal direction is

  (6)

  Since this velocity is constant, the horizontal distance x traveled in time t is

  (7)

  The vertical motion is slightly more complicated. If there were no gravitational force pulling the projectile downward, it would continue to travel upward at the constant velocity V sin A . However, as the projectile rises gravity pulls it down. Now we saw in chapter 12 that any object subject to the action of gravity will have an acceleration in the downward direction of 32 feet per second each second and will therefore acquire a downward velocity of 32t feet in t seconds. Hence the first statement we may make about the vertical velocity of the projectile is that its net velocity v y at any time t is

  (8)

  How far above the ground will the projectile be any time t? Well, the velocity V sin A will cause the projectile to rise Vt sin A feet in t seconds. During these t seconds the action of gravity will pull the projectile downward 16t 2 feet. Hence the net vertical distance y attained by the projectile in t seconds is

  (9)

  In deriving formulas (8) and (9) we have assumed that the motions of rising and falling may be considered separately, even though they take place simultaneously. But each motion is due to a separate and independent force, just as the horizontal and vertical motions are independent, and, consequently, our derivation rests on a sound physical principle.

  Formulas (7) and (9) describe the separate motions in the horizontal and vertical directions, each giving a distance in terms of time t . For any given time t we could calculate x and y and thus know where the projectile is at that time. However, we might like to know what the actual path of the projectile is without having to calculate successive x and y

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  Sports Math

  An Introductory Course in the Mathematics of Sports Science and Sports Analytics

  • Roland B. Minton(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (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 Lionel 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 questions 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 fundamentals 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 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. At a constant speed, this means that velocity equals change in height divided by change in time:

  υ =

  Δ h

  Δ t

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