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
5
4
Maximum height (m)
This scaled
diagram shows
the size and
shape of
trajectories for
an object
projected at 10
m/s at different
angles.
3
2
1
0
0
1
2
3
4
5
6
Range (distance) (m)
7 8
9 10 11
Factors Influencing Projectile
Trajectory
The Effect of Projection Angle on Range
(Relative Projection Height = 0)
Projection
Speed
(m/s)
Projection
Angle
(degrees)
10
10
10
10
10
10
10
10
10
10
20
30
40
45
50
60
70
80
Range
(m)
3.49
6.55
8.83
10.04
10.19
10.04
8.83
6.55
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 velocity
Relative projection height
Horizontal displacement
Horizontal velocity
Initial vertical velocity
Vertical displacement
Relative projection height
Initial vertical velocity
Trajectory
Initial speed
Projection 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
Standing Broad
Jump take-off
P2
P1
• 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
+
-
+
-
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
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.
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.
apex
gravity
Vertical velocity
decreases as the
ball rises and
increases as the
ball falls due to
the influence of
gravitational
force.
Calculation of Displacement
• Calculation of Magnitude:
P2
P1
Resultant displacement (dR)
=
=
= 0.63 m
• Calculation of Direction:
Resultant
displacement
(dR)

Horizontal
displacement (dH) =
0.6 m
Vertical
displacement
(dV) = 0.2 m
Angle to horizontal (θ)
Tan θ = Opposite / Adjacent
Tan θ = dV / dH = 0.2 / 0.6
θ = Tan-1 (0.2 / 0.6)
θ = 18.8º
Calculation of components of velocity
At take-off in SBJ
θ
Vertical component of
velocity (vV)
Horizontal component of
velocity (vH)
vR = 3.2 m·s-1
θ = 23º
Horizontal 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-1
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 = v12 + 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 = v12 + 2·0·d
If applied to vertical projectile free falling (v1 =0),
1. v2 = v1 (0) + at
2. d = v1 (0) t + ½ at2
3. v22 = v12 (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