Presentation on theme: "Kinematics of Projectile Motion"— Presentation transcript:
1Kinematics of Projectile Motion What is a projectile?A body in free fall that is subject only to the forces of gravity and air resistanceMotion of bodies flung into the airOccurs in many activities, such as baseball, diving, figure skating, basketball, golf, and volleyballA special case of linear kinematics
2Kinematics of Projectile Motion Projectiles have different objectivesTime of flightMaximum – tennis defensive lob, football punt, springboard diving, ski/snowboard ariel tennis lobMinimum – baseball infield throw, tennis volleyMaximum 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
3Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile?projection angle - the direction of projection with respect to the horizontal
4Factors 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.
5Factors Influencing Projectile Trajectory Maximum height (m)Range (distance) (m)54321This scaled diagram shows the size and shape of trajectories for an object projected at 10 m/s at different angles.
6Factors Influencing Projectile Trajectory The Effect of Projection Angle on Range(Relative Projection Height = 0)Projection ProjectionSpeed Angle Range(m/s) (degrees) (m)
7Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile?projection speed - the magnitude of projection velocity
8Factors 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.
9Factors Influencing Projectile Trajectory What factors influence the trajectory (flight path) of a projectile?relative projection height - the difference between projection height and landing height
10Factors 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.
12Generalizations for Maximum Range If purpose to maximize range, optimum angle of landing is always 45º. 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°.
13Standing Broad Jump take-off Projectile as a VectorInitial velocity of projectile is a vectorSpeed (Magnitude)Angle (Direction)Point of originVector represented graphically by:Line of actionInitial velocity of projectile resolved into horizontal and vertical componentsIf horizontal and vertical components added, resultant equals original initial velocityStanding Broad Jump take-offP2P1+-
14Vector 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)
15Vector Components of Projectile Motion Horizontal component (Vh) has certain velocity or magnitude.Horizontal component (Vh) remains constant throughout flight, neglecting air resistance.Horizontal velocity influences range, but not time object in air.
16Kinematics of Projectile Motion Downward acceleration of a projectile same as downward acceleration of a free falling body due to constant gravity.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.
17Kinematics of Projectile Motion Horizontal velocity (Vh) does not affect vertical velocity (Vv).(Vh) and (Vv) are independent of one anotherGravity affects vertical velocity (Vv).What is the effect of gravity?(The force of gravity produces a constant acceleration of m/s2 or ft/s2 on bodies near the surface of the earth.)Negative (-) vertical direction is downward.
18Kinematics of Projectile Motion apexgravityThe 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.
19Calculation of Displacement Calculation of Magnitude:Resultant displacement (dR)== mCalculation of Direction:Angle to horizontal (θ)Tan θ = Opposite / AdjacentTan θ = dV / dH = 0.2 / 0.6θ = Tan-1 (0.2 / 0.6)θ = 18.8ºP1P2Vertical displacement (dV) = 0.2 mHorizontal displacement (dH) = 0.6 mResultant displacement (dR)
20Calculation of components of velocity vR = 3.2 m·s-1θ = 23ºAt take-off in SBJθHorizontal component of velocity (vH): cos θ = Adjacent / Hypotenuse cos θ = vH / vR vH = vR × cos θ vH = 3.2 × cos 23 vH = 2.94 m·s-1 Vertical component of velocity (vV): sin θ = Opposite / Hypotenuse sin θ = vV / vR vV = vR × sin θ vV = 3.2 × sin 23 vV = 1.25 m·s-1Vertical component of velocity (vV)Horizontal component of velocity (vH)
21Equations of Constant Acceleration Three formulas interrelating the kinematic quantities – displacement, velocity, acceleration, and time.v2 = v1 + atd = v1t + ½ at2v22 = v12 + 2adThe equation that you select to solve a problem must have the known quantities and the unknown variable you wish to find.
22Equations of Constant Acceleration If applied to horizontal projectile in which a = 0,v2 = v1 + 0·td = v1t + ½ 0·t2v22 = v12 + 2·0·dIf applied to vertical projectile free falling (v1 =0),v2 = v1 (0) + atd = v1 (0) t + ½ at2v22 = v12 (0) + 2ad
23Summary 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 bodyAverage velocity is the change in position divided by the change in timeAverage acceleration is the change in velocity divided by the change in timeThe resultant and angle of a vector variable can be calculated from its horizontal and vertical components using Pythagorean Theorem and trigonometryThe horizontal and vertical components of a vector variable can be calculated from its resultant and angle using trigonometry