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Projectile Motion or 2D Kinematics By Sandrine Colson-Inam, Ph.D References: Conceptual Physics, Paul G. Hewitt, 10 th edition, Addison Wesley publisher http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3 l2a.html http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3 l2a.html

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Outline What is a projectile Characteristics of a projectile's motion Horizontal and vertical components of velocity and displacement Initial velocity components Examples of problems

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What is a projectile? A projectile is an object upon which the only force acting is gravity. A projectile is any object which once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

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Projectile Motion and Inertia

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Horizontal and Vertical Velocities A projectile is any object upon which the only force is gravity, Projectiles travel with a parabolic trajectory due to the influence of gravity, There are no horizontal forces acting upon projectiles and thus no horizontal acceleration, The horizontal velocity of a projectile is constant (a never changing value), There is a vertical acceleration caused by gravity; its value is - 9.8 m/s/s, or DOWNWARD!! The vertical velocity of a projectile changes by 9.8 m/s each second, The horizontal motion of a projectile is independent of its vertical motion.

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Vector diagrams for projectile motion

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Horizontal and vertical displacement – Horizontally Launched Projectile y = 0.5 g t 2 (equation for vertical displacement for a horizontally launched projectile) If the horizontal displacement (x) of a projectile were represented by an equation, then that equation would be written as x = v ix t

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Displacement diagram of projectile motion HorizontalVertical

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Horizontal and vertical displacement : Non - Horizontally Launched Projectile y = v iy t + 0.5 g t 2 (equation for vertical displacement for an angled-launched projectile) If the horizontal displacement (x) of a projectile were represented by an equation, then that equation would be written as x = v ix t

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Check your understanding 1. Anna Litical drops a ball from rest from the top of 78.4-meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of motion? 2. A cannonball is launched horizontally from the top of an 78.4-meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of travel?

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Answers 1. It will take 4 seconds to fall 78.4 meters Use the equation y = 0.5 g t 2 and substitute -9.8 m/s/s for g. The vertical displacement must then be subtracted from the initial height of 78. 4 m. At t = 1 s, y = 4.9 m (down) so height is 73.5 m (78.4 m - 4.9 m ) At t = 2 s, y = 19.6 m (down) so height is 58.8 m (78.4 m - 19.6 m ) At t = 3 s, y = 44.1 m (down) so height is 34.3 m (78.4 m - 45 m) At t = 4 s, y = 78.4 m (down) so height is 0 m (78.4 m - 78.4 m) 2. It will take 4 seconds to fall 78.4 meters Use the equation y = 0.5 g t 2 and substitute -9.8 m/s/s for g. The vertical displacement must then be subtracted from the initial height of 78. 4 m. At t = 1 s, y = 4.9 m (down) so height is 73.5 m (78.4 m - 4.9 m ) At t = 2 s, y = 19.6 m (down) so height is 58.8 m (78.4 m - 19.6 m ) At t = 3 s, y = 44.1 m (down) so height is 34.3 m (78.4 m - 45 m) At t = 4 s, y = 78.4 m (down) so height is 0 m (78.4 m - 78.4 m)

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Check your understanding 3. Fill in the table indicating the value of the horizontal and vertical components of velocity and acceleration for a projectile. 4. The diagram below shows the trajectory for a projectile launched non-horizontally from an elevated position on top of a cliff. The initial horizontal and vertical components of the velocity are 8 m/s and 19.6 m/s respectively. Positions of the object at 1-second intervals are shown. Determine the horizontal and vertical velocities at each instant shown in the diagram.

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Answers 3. The v x values will remain constant at 15.0 m/s for the entire 6 seconds; the a x values will be 0 m/s/s for the entire 6 seconds. The v y values will be changing by -9.8 m/s each second. Thus, v y = 29.4 m/s (t = 0 s) v y = 19.6 m/s (t = 1 s) v y = 9.8 m/s (t = 2 s)v y = 0 m/s (t = 3 s) v y = -9.8 m/s (t = 4 s)v y = -19.6 m/s (t = 5 s) v y = -29.4 m/s (t = 6 s) The a y values will be -9.8 m/s/s for the entire 6 seconds. 4. The v x values will remain 8 m/s for the entire 6 seconds. The v y values will be changing by 9.8 m/s each second. Thus, v y =9.8 m/s (t = 1 s) v y = 0 m/s (t = 2 s) v y = -9.8 m/s (t = 3 s)v y = -19.6 m/s (t = 4 s) v y = -29.4 m/s (t = 5 s)v y = -39.2 m/s (t = 6 s)

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Initial components of velocity v x = v cos α ° v y = v sin α °

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Evaluating various info Determination of the Time of Flight Determination of Horizontal Displacement x = v ix t Determination of the Peak Height y = v iy t + 0.5 g t 2

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Equations of motion Horizontal motion Vertical Motion the vertical acceleration of a projectile is known to be -9.8 m/s/s

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Solving Projectile Motion Problems The following procedure summarizes the above problem-solving approach. Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. For convenience sake, make a table with horizontal information on one side and vertical information on the other side. Identify the unknown quantity which the problem requests you to solve for. Select either a horizontal or vertical equation to solve for the time of flight of the projectile. With the time determined, use one of the other equations to solve for the unknown. (Usually, if a horizontal equation is used to solve for time, then a vertical equation can be used to solve for the final unknown quantity.)

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Check your understanding A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal displacement, and the peak height of the football.

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Answer State the problem Use the appropriate equations of motion The unknown quantities are the horizontal displacement, the time of flight, and the height of the football at its peak. From the vertical information in the table above and the second equation listed among the vertical kinematic equations (v fy = v iy + a y *t), it becomes obvious that the time of flight of the projectile can be determined. By substitution of known values, the equation takes the form of-17.7 m/s = 17.7 m/s + (-9.8 m/s/s)tvertical kinematic equations -35.4 m/s = (-9.8 m/s/s)t 3.61 s = t The total time of flight of the football is 3.61 seconds. With the time determined, information in the table and the horizontal kinematic equations can be used to determine the horizontal displacement (x) of the projectile. The first equation (x = v ix t + 0.5a x t 2 ) listed among the horizontal kinematic equations is suitable for determining x. With the equation selected, the physics problem once more becomes transformed into an algebra problem. By substitution of known values, the equation takes the form ofx = (17.7 m/s)(3.6077 s) + 0.5(0 m/s/s)(3.6077 s) 2horizontal kinematic equations x = (17.7 m/s)(3.6077 s) x = 63.8 m The horizontal displacement of the projectile is 63.8 m. Finally, the problem statement asks for the height of the projectile at is peak. This is the same as asking, "what is the vertical displacement (y) of the projectile when it is halfway through its trajectory?" In other words, find y when t = 1.80 seconds (one-half of the total time). To determine the peak height of the projectile (y with t = 1.80 sec), the first equation (y = v iy t +0.5a y t 2 ) listed among the vertical kinematic equations can be used. By substitution of known values into this equation, it takes the form ofproblem statementthe total timevertical kinematic equations y = (17.7 m/s)(1.80 s) + 0.5*(-10 m/s/s)(1.80 s) 2 y = 31.9 m + (-15.9 m) y = 15.9 m The solution to the problem statement yields the following answers: the time of flight of the football is 3.61 s, the horizontal displacement of the football is 63.8 m, and the peak height of the football 15.9 m.problem statement

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The Problem-Solving Approach The following procedure summarizes the above problem-solving approach. Use the given values of the initial velocity (the magnitude and the angle) to determine the horizontal and vertical components of the velocity (v ix and v iy ). Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. For convenience sake, make a table with horizontal information on one side and vertical information on the other side. Identify the unknown quantity which the problem requests you to solve for. Select either a horizontal or vertical equation to solve for the time of flight of the projectile. For non-horizontally launched projectiles, the second equation listed among the vertical equations (v fy = v iy + a y *t) is usually the most useful equation.vertical equations With the time determined, use a horizontal equation (usually x = v ix *t + 0.5*a x *t 2 ) to determine the horizontal displacement of the projectile. Finally, the peak height of the projectile can be found using a time value which one- half the total time of flight. The most useful equation for this is usually y = v iy *t +0.5*a y *t 2.

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