Velocity of a Projectile

11 Key excerpts on "Velocity of a Projectile"

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  Physics

  • 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.

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  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.

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  Physics

  • 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.

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  Physics

  • 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.

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  Introduction to Physics

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

    (Publisher)

  7. Keep in mind that a kinematics problem may have two possible answers. Try to visualize the different physical situations to which the answers correspond. Check Your Understanding (The answer is given at the end of the book.) 2. A power boat, starting from rest, maintains a constant acceleration. After a certain time t, its displacement and velocity are r B and v B . At time 2t, what would be its displacement and veloc- ity, assuming the acceleration remains the same? (a) 2 r B and 2 v B (b) 2 r B and 4 v B (c) 4 r B and 2 v B (d) 4 r B and 4 v B 3.3 | Projectile Motion The biggest thrill in baseball is a home run. The motion of the ball on its curving path into the stands is a common type of two-dimensional motion called “projectile motion.” A good description of such motion can often be obtained with the assumption that air resistance is absent. Using the equations in Table 3.1, we consider the horizontal and vertical parts of the motion separately. In the horizontal or x direction, the moving object (the projectile) does not slow down in the absence of air resistance. Thus, the x component of the velocity re- mains constant at its initial value or v x 5 v 0x , and the x component of the acceleration is a x 5 0 m/s 2 . In the vertical or y direction, however, the projectile experiences the effect of gravity. As a result, the y component of the velocity v y is not constant and changes. The y component of the acceleration a y is the downward acceleration due to gravity. If the path or trajectory of the projectile is near the earth’s surface, a y has a magnitude of 9.80 m/s 2 . In this text, then, the phrase “projectile motion” means that a x 5 0 m/s 2 and a y equals the acceleration due to gravity. Example 3 and other examples in this section illustrate how the equations of kinematics are applied to projectile motion. EXAMPLE 3 | A Falling Care Package Figure 3.7 shows an airplane moving horizontally with a constant velocity of 1115 m/s at an altitude of 1050 m.

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  Fundamentals of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  (4.3.2) In unit-vector notation, a → = a x ˆ i + a y ˆ j + a z k ̂ , (4.3.3) where a x = dv x /dt, a y = dv y /dt, and a z = dv z /dt. Review & Summary Projectile Motion Projectile motion is the motion of a par- ticle that is launched with an initial velocity v → 0 . During its flight, the particle’s horizontal acceleration is zero and its vertical acceleration is the free-fall acceleration ‒g. (Upward is taken to be a positive direction.) If v → 0 is expressed as a magnitude (the speed v 0 ) and an angle θ 0 (measured from the horizontal), the particle’s equations of motion along the horizontal x axis and vertical y axis are x − x 0 = ( v 0 cos θ 0 )t, (4.4.3) y − y 0 = ( v 0 sin θ 0 )t − 1 _ 2 gt 2 , (4.4.4) v y = v 0 sin θ 0 − gt, (4.4.5) v y 2 = ( v 0 sin θ 0 ) 2 − 2g(y − y 0 ). (4.4.6) The trajectory (path) of a particle in projectile motion is parabolic and is given by y = (tan θ 0 )x − gx 2 ____________ 2( v 0 cos θ 0 ) 2 , (4.4.7) if x 0 and y 0 of Eqs. 4.4.3 to 4.4.6 are zero. The particle’s horizontal range R, which is the horizontal distance from the launch point to the point at which the particle returns to the launch height, is R = v 0 2 __ g sin 2θ 0 . (4.4.8) Uniform Circular Motion If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a → of constant magnitude a = v 2 ___ r . (4.5.1) The direction of a → is toward the center of the circle or circular arc, and a → is said to be centripetal. The time for the particle to complete a circle is T = 2πr ____ v . (4.5.2) T is called the period of revolution, or simply the period, of the motion. Relative Motion When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B.

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  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).

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  velocity of plane relative to ground (PG) velocity of plane relative to wind (PW) velocity of wind relative to ground. (WG) = + 73 Projectile Motion Projectile motion is the motion of a particle that is launched with an initial velocity v → 0 . During its flight, the par- ticle’s horizontal acceleration is zero and its vertical acceleration is the free-fall acceleration ‒g. (Upward is taken to be a positive direction.) If v → 0 is expressed as a magnitude (the speed v 0 ) and an angle θ 0 (measured from the horizontal), the particle’s equations of motion along the horizontal x axis and vertical y axis are x − x 0 = (v 0 cos θ 0 )t, (4-21) y − y 0 = (v 0 sin θ 0 )t − 1 2 gt 2 , (4-22) v y = v 0 sin θ 0 − gt, (4-23) v 2 y = (v 0 sin θ 0 ) 2 − 2g(y − y 0 ). (4-24) The trajectory (path) of a particle in projectile motion is parabolic and is given by y = (tan θ 0 )x − gx 2 2(v 0 cos θ 0 ) 2 , (4-25) if x 0 and y 0 of Eqs. 4-21 to 4-24 are zero. The particle’s horizontal range R, which is the horizontal distance from the launch point to the point at which the particle returns to the launch height, is R = v 2 0 g sin 2θ 0 . (4-26) Uniform Circular Motion If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a → of constant magnitude a = v 2 r . (4-34) The direction of a → is toward the center of the circle or circular arc, and a → is said to be centripetal. The time for the particle to complete a circle is T = 2πr v . (4-35) T is called the period of revolution, or simply the period, of the motion. Relative Motion When two frames of reference A and B are mov- ing relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B. The two measured velocities are related by v → PA = v → PB + v → BA , (4-44) where v → BA is the velocity of B with respect to A.

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  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.

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  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.

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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)
  • CRC Press

    (Publisher)

  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: v = Δ h Δ t . This gets complicated when velocity is not constant. In general, Average velocity = Δ h Δ t and, for small time intervals, (instantaneous) velocity is approximately equal to average velocity: v ≈ Δ h Δ t . With calculus, we can simply say that velocity is the derivative of height. Either way, note that v can be negative (if height is decreasing) or positive (if height is increasing). The acceleration a of the object is, in turn, the rate of change of velocity. Then a ≈ Δ v Δ t and acceleration is the derivative of velocity. Example 1.1 Suppose a ball falls from a height of 50 meters. If gravity is the only force on the ball, find the velocity of the ball after t = 1 second and t = 1 . 5 seconds. Solution . For most sports situations, we can assume that the acceleration due to gravity is a constant -g with g ≈ 9 . 8 m/s 2 or g ≈ 32 ft/s 2 . An acceleration of 9 . 8 m/s 2 in the negative direction means that in every second the velocity decreases by 9 . 8 m/s. Assuming that the ball starts with velocity 0, then at t = 1 second the velocity has decreased to -9 . 8 m/s. In the next half-second, the velocity decreases by 0 . 5(9 . 8) m/s = 4 . 9 m/s. At time t = 1 . 5 s the velocity has decreased to ( -9 . 8 -4 . 9) m/s = -14 . 7 m/s. The ideas from this basic example will be used again for the more complicated situation of Figure 1.9. Speed is defined as the absolute value of velocity. In Example 1.1 above, at time t = 1 the ball’s velocity is -9 . 8 m/s but its speed is 9 . 8 m/s (downward). Notice that Example 1.1 did not ask for heights.

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