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

Kinematics of Projectile Motion

• Projectiles have different objectives– Time of flight

• Maximum – tennis defensive lob, football punt, springboard diving, ski/snowboard ariel tennis 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

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

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.

Factors Influencing Projectile Trajectory

Max

imum

hei

ght

(m)

Range (distance) (m)0 1 2 3 4 5 6 7 8 9 10 11

5

4

3

2

1

0

This scaled diagram shows the size and shape of trajectories for an object projected at 10 m/s at different angles.

Factors Influencing Projectile Trajectory

The 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 3.49

Factors Influencing Projectile Trajectory

What factors influence the trajectory (flight path) of a projectile?

• projection speed - the magnitude of projection velocity

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.

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

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.

Factors Influencing Projectile Trajectory

FACTORS INFLUENCING PROJECTILE MOTION(Neglecting Air Resistance)

Variable Factors of Influence

Flight time Initial vertical velocityRelative projection height

Horizontal displacement Horizontal velocityInitial vertical velocity

Relative projection heightVertical displacementInitial vertical velocity

Trajectory Initial speedProjection angle

Relative projection height

Generalizations 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°.

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

+-

+

-

P2P1

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)

Vector 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.

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.

Kinematics of Projectile Motion

• Horizontal velocity (Vh) does not affect vertical velocity (Vv).

• (Vh) and (Vv) are independent of one another

• Gravity 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 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.

apex

gravity

Calculation of Displacement

22 )2.0(+)6.0(2

V2

H d+d

• Calculation of Magnitude:Resultant displacement (dR)

==

= 0.63 m

• Calculation of Direction:Angle to horizontal (θ)

Tan θ = Opposite / AdjacentTan θ = dV / dH = 0.2 / 0.6

θ = Tan-1 (0.2 / 0.6) θ = 18.8º

Vertical displacement (dV) = 0.2 m

Horizontal displacement (dH) = 0.6 m

Resultant displacement (dR)

P1

P2

Calculation of components of velocity

Horizontal component of velocity (vH):

cos θ = Adjacent / Hypotenuse cos θ = vH / vRvH = vR × cos θvH = 3.2 × cos 23 vH = 2.94 m·s-1

Vertical component of velocity (vV):

sin θ = Opposite / Hypotenuse sin θ = vV / vRvV = vR × sin θvV = 3.2 × sin 23vV = 1.25 m·s-1

vR = 3.2 m·s-1

θ = 23º

At take-off in SBJ

θ

Horizontal component of velocity (vH)

Vertical component of velocity (vV)

Equations of Constant Acceleration

Three formulas interrelating the kinematic quantities – displacement, velocity, acceleration, and time.

1. v2 = v1 + at

2. d = v1t + ½ at2

3. v22 = v1

2 + 2ad

The equation that you select to solve a problem must have the known quantities and the unknown variable you wish to find.

Equations of Constant Acceleration

If applied to horizontal projectile in which a = 0,1. v2 = v1 + 0·t

2. d = v1t + ½ 0·t2

3. v22 = v1

2 + 2·0·d

If applied to vertical projectile free falling (v1 =0),

4. v2 = v1 (0) + at

5. d = v1 (0) t + ½ at2

6. v22 = v1

2 (0) + 2ad

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