Kinematics of Projectile Motion

Kinematics of Projectile MotionWhat 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 kinematicsKinematics of Projectile MotionProjectiles 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

Factors Influencing Projectile TrajectoryWhat factors influence the trajectory (flight path) of a projectile? projection angle - the direction of projection with respect to the horizontalFactors 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.Factors Influencing Projectile TrajectoryMaximum height (m)Range (distance) (m)0 1 2 3 4 5 6 7 8 9 10 11543210This scaled diagram shows the size and shape of trajectories for an object projected at 10 m/s at different angles.Factors Influencing Projectile TrajectoryThe Effect of Projection Angle on Range(Relative Projection Height = 0)Projection 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 3.49Factors Influencing Projectile TrajectoryWhat factors influence the trajectory (flight path) of a projectile? projection speed - the magnitude of projection velocityFactors Influencing Projectile TrajectoryWhen 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.

Factors Influencing Projectile TrajectoryWhat factors influence the trajectory (flight path) of a projectile? relative projection height - the difference between projection height and landing heightFactors 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.Factors Influencing Projectile TrajectoryFACTORS INFLUENCING PROJECTILE MOTION(Neglecting Air Resistance)Variable Factors of InfluenceFlight timeInitial vertical velocityRelative projection heightHorizontal displacementHorizontal velocityInitial vertical velocityRelative projection heightVertical displacementInitial vertical velocityTrajectoryInitial speedProjection angleRelative projection height11Generalizations for Maximum RangeIf purpose to maximize range, optimum angle of landing is always 45.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.

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-off+-+-P2P113Vector Components of Projectile MotionWhy 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)Vector Components of Projectile MotionHorizontal 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.

Kinematics of Projectile MotionTwo 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.Kinematics of Projectile MotionHorizontal 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 -9.81 m/s2 or -32.2 ft/s2 on bodies near the surface of the earth.)Negative (-) vertical direction is downward.Kinematics of Projectile MotionThe 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.apexgravityCalculation of Displacement

Calculation of Magnitude:Resultant displacement (dR)

=

=

= 0.63 m

Calculation of Direction:

Angle to horizontal ()

Tan = Opposite / Adjacent

Tan = dV / dH = 0.2 / 0.6

= Tan-1 (0.2 / 0.6)

= 18.8Vertical displacement (dV) = 0.2 mHorizontal displacement (dH) = 0.6 mResultant displacement (dR)P1P219Calculation of components of velocityHorizontal component of velocity (vH):

cos = Adjacent / Hypotenuse cos = vH / vRvH = vR cos vH = 3.2 cos 23 vH = 2.94 ms-1

Vertical component of velocity (vV):

sin = Opposite / Hypotenuse sin = vV / vRvV = vR sin vV = 3.2 sin 23vV = 1.25 ms-1vR = 3.2 ms-1 = 23At take-off in SBJHorizontal component of velocity (vH)Vertical component of velocity (vV)20Equations of Constant AccelerationThree formulas interrelating the kinematic quantities displacement, velocity, acceleration, and time.v2 = v1 + atd = v1t + at2v22 = v12 + 2adThe equation that you select to solve a problem must have the known quantities and the unknown variable you wish to find. 21Equations of Constant AccelerationIf applied to horizontal projectile in which a = 0,v2 = v1 + 0td = v1t + 0t2v22 = v12 + 20dIf applied to vertical projectile free falling (v1 =0),v2 = v1 (0) + atd = v1 (0) t + at2v22 = v12 (0) + 2adSummaryVariables 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 trigonometry23