Velocity of a Projectile

4 Key excerpts on "Velocity of a Projectile"

• 

  eBook - ePub

  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

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

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

    (Publisher)

  Chapter 4

  Projectile Motion

  The 1-dimensional kinematics of Chapter 2 allow us to describe horizontal or vertical motion in a straight line. Vectors allow us to analyze 2-dimensional objects. Now we combine the two and use vectors to describe 2-dimensional motion. A 2-d kinematics problem is really two 1-d kinematics problems and we use vectors to break the analysis into components.

  One example of 2-d motion is projectile motion, where an object is launched with an initial velocity and then is influenced only by gravity and air resistance. The object’s initial coordinates are x0 and y0 . Its initial velocity has magnitude v0 and direction θ0 above the horizontal.

  The horizontal component of the initial velocity is vox = v0 cos θ0 and the vertical component is voy = v0 sin θ0 . This unusual notation is due to a limitation of SNB: the engine can’t handle a double subscript unless the indices are either both numbers or both letters. So we’ll use vox instead of vox and voy instead of v

  0y

  . Here is a summary of the notation for the velocity and position of our projectile.

  Table 4.1

  With the equations of 1-dimensional kinematics and vectors, we can analyze projectile motion with and without air resistance. We will calculate the horizontal position x and the height 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)

  Sign up to read

  Learn more about book
• 

  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

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  BIOS Instant Notes in Sport and Exercise Biomechanics

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

    (Publisher)

  Fig. B5.5.

  Considering

  Fig. B5.5

  in terms of the vertical component of the ball’s motion, we can see that it travels upward and downward (displacement/time graph;

  Fig. B5.5

  : graph 1) with a decreasing vertical velocity (positive value) as it travels upward. The ball then reaches the peak height of the flight path and the velocity changes direction (i.e., it stops going upward and instantly starts coming downward) and throughout this action it has been accelerating at a constant rate (–9.81 m/s2 ) with a decreasing positive vertical velocity and an increasing negative vertical velocity (graphs 2 and 3). This is exactly the same as when the ball that was thrown perfectly vertically (providing the vertical release velocity was the same in both experiments). Horizontally, the ball will be displaced as shown

  Fig. B5.5

  : graph 4. It will travel forwards with constant horizontal velocity (graph 5) in accordance with Newton’s first law and it will do so with zero horizontal acceleration (constant velocity horizontally as in graphs 5 and 6). Hence, vertical and horizontal motions during projectile flight are independent of each other and gravity affects the vertical component only.

  Fig. B5.5. The graphical representation of the motion of a ball thrown with vertical and horizontal velocity

  Air resistance

  In the understanding of vertical projection it is worth making a comment about the effects of air resistance. Normally, in human motion we consider the effects of air resistance to be negligible (particularly on the human body as it travels as a projectile through the air). However, in certain applications the effects of air resistance will not be negligible and will be considered as an external force that affects motion. For example, in the case of dropping objects vertically, we know from Newton’s law of gravitation that any object near to or on its surface regardless of its mass will accelerate toward the ground at a constant rate (i.e., two objects of different masses when dropped at the same height will both hit the ground at the same time). However, if you take the case of dropping a piece of paper and a golf ball you will see that the golf ball will hit the ground first. In this case air resistance will affect the piece of paper by a significant amount such that its descent towards the Earth will be slowed down (air resistance becomes an external force). Similarly, in sports such as javelin, hammer throwing, and discus, and even to an extent in long jumping when there are “head and tail” winds air resistance will have an effect. Often long jumps that are wind assisted are not legitimate jumps (in this case the tail wind would be an external force of assistance). Hence, in certain sports and movements it may be the case that the air resistance effects should be considered to be more than negligible. Experiment with dropping different objects from the same height to see if you can demonstrate the effects of air resistance on the vertical downward acceleration of objects caused by the force of gravity

  Sign up to read

  Learn more about book