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Kinematics of Projectile Motion What is a projectile? – A body in free fall that is subject only to the forces of gravity and air resistance – Motion of bodies flung into the air – Occurs in many activities, such as baseball, diving, figure skating, basketball, golf, and volleyball – A special case of linear kinematics

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Kinematics of Projectile Motion Projectiles have different objectives – Time of flight Maximum – tennis defensive lob, football punt, springboard diving, ski/snowboard ariel tennis lobtennis lob Minimum – baseball infield throw, tennis volley – Maximum horizontal displacement (range) - javelin, discus, shot put, long jump, triple jump, football kickoff, golf drive, – Maximum vertical displacement (apex) – pole vault, high jump, basketball jump ball

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Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile? projection angle - the direction of projection with respect to the horizontal

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Factors Influencing Projectile Trajectory Trajectory shape dependent on angle of projection in absence of air resistance. If angle perfectly vertical, trajectory also vertical. If angle oblique, trajectory is parabolic. If angle horizontal, trajectory is half parabola.

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Factors Influencing Projectile Trajectory Maximum height (m) Range (distance) (m) 0 1 2 3 4 5 6 7 8 9 10 11 543210543210 This scaled diagram shows the size and shape of trajectories for an object projected at 10 m/s at different angles.

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Factors Influencing Projectile Trajectory The Effect of Projection Angle on Range (Relative Projection Height = 0) Projection Speed Angle Range (m/s)(degrees) (m) 10 10 3.49 10 20 6.55 10 30 8.83 10 40 10.04 10 45 10.19 10 50 10.04 10 60 8.83 10 70 6.55 10 80 3.49

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Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile? projection speed - the magnitude of projection velocity

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Factors Influencing Projectile Trajectory When projection angle and other factors constant, projection speed determines length of trajectory (range). For vertical projectile, speed determines apex. For oblique projectile, speed determines height of apex and horizontal range.

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Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile? relative projection height - the difference between projection height and landing height

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Factors Influencing Projectile Trajectory When projection speed is constant, greater relative projection height provides longer flight time which increases horizontal displacement. Taller shot putters can throw farther than shorter ones even if throw with same speed.

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Factors Influencing Projectile Trajectory FACTORS INFLUENCING PROJECTILE MOTION (Neglecting Air Resistance) Variable Factors of Influence Flight time Initial vertical velocity Relative projection height Horizontal displacement Horizontal velocity Initial vertical velocity Relative projection heightVertical displacement Initial vertical velocity Trajectory Initial speed Projection angle Relative projection height

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Generalizations for Maximum Range If purpose to maximize range, optimum angle of landing is always 45 º. If purpose to maximize range & projection height is zero, the optimum angle of projection (and landing) is 45°. If purpose to maximize range & projection height is above landing (+), optimum angle of projection less than 45°.

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Projectile as a Vector Initial velocity of projectile is a vector – Speed (Magnitude) – Angle (Direction) – Point of origin Vector represented graphically by: – Line of action Initial velocity of projectile resolved into horizontal and vertical components – If horizontal and vertical components added, resultant equals original initial velocity Standing Broad Jump take-off + - + - P2 P1

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Vector Components of Projectile Motion Why do we analyze the horizontal and vertical components of projectile motion separately? (the vertical component is influenced by gravity and the horizontal component is not)

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Vector Components of Projectile Motion Horizontal component (V h ) has certain velocity or magnitude. Horizontal component (V h ) remains constant throughout flight, neglecting air resistance. Horizontal velocity influences range, but not time object in air.

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Kinematics of Projectile Motion Two balls - one dropped and one projected horizontally from the same height: Both land at the same time since gravity affects their vertical velocities equally. Downward acceleration of a projectile same as downward acceleration of a free falling body due to constant gravity.

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Kinematics of Projectile Motion Horizontal velocity (V h ) does not affect vertical velocity (V v ). (V h ) and (V v ) are independent of one another Gravity affects vertical velocity (V v ). What is the effect of gravity? –(The force of gravity produces a constant acceleration of -9.81 m/s 2 or -32.2 ft/s 2 on bodies near the surface of the earth.) –Negative (-) vertical direction is downward.

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Kinematics of Projectile Motion The pattern of change in the vertical velocity of a projectile is symmetrical about the apex. Vertical velocity decreases as the ball rises and increases as the ball falls due to the influence of gravitational force. apexgravity

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Calculation of Displacement Calculation of Magnitude: Resultant displacement (d R ) = = = 0.63 m Calculation of Direction: Angle to horizontal (θ) Tan θ = Opposite / Adjacent Tan θ = d V / d H = 0.2 / 0.6 θ = Tan -1 (0.2 / 0.6) θ = 18.8º Vertical displacement (d V ) = 0.2 m Horizontal displacement (d H ) = 0.6 m Resultant displacement (d R ) P1 P2

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Calculation of components of velocity Horizontal component of velocity (v H ): cos θ = Adjacent / Hypotenuse cos θ = v H / v R v H = v R × cos θ v H = 3.2 × cos 23 v H = 2.94 m·s -1 Vertical component of velocity (v V ): sin θ = Opposite / Hypotenuse sin θ = v V / v R v V = v R × sin θ v V = 3.2 × sin 23 v V = 1.25 m·s -1 v R = 3.2 m·s -1 θ = 23º At take-off in SBJ θ Horizontal component of velocity (v H ) Vertical component of velocity (v V )

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Equations of Constant Acceleration Three formulas interrelating the kinematic quantities – displacement, velocity, acceleration, and time. 1.v 2 = v 1 + at 2.d = v 1 t + ½ at 2 3.v 2 2 = v 1 2 + 2ad The equation that you select to solve a problem must have the known quantities and the unknown variable you wish to find.

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Equations of Constant Acceleration If applied to horizontal projectile in which a = 0, 1.v 2 = v 1 + 0·t 2.d = v 1 t + ½ 0·t 2 3.v 2 2 = v 1 2 + 2·0·d If applied to vertical projectile free falling (v 1 =0), 1.v 2 = v 1 (0) + at 2.d = v 1 (0) t + ½ at 2 3.v 2 2 = v 1 2 (0) + 2ad

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Summary Variables used to describe motion are either: – Scalar (magnitude only: e.g. time, distance and speed) – Vector (magnitude and direction: e.g. displacement, velocity and acceleration) Displacement is the change in position of a body Average velocity is the change in position divided by the change in time Average acceleration is the change in velocity divided by the change in time The resultant and angle of a vector variable can be calculated from its horizontal and vertical components using Pythagorean Theorem and trigonometry The horizontal and vertical components of a vector variable can be calculated from its resultant and angle using trigonometry

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