Projectile Motion
• Concerns the flight of an object or body
after it is free of support. (This includes
objects that are dropped.)
• The flight path of a projectile is called the
trajectory.
• Objects that are continuously being
propelled (such as airplanes) aren’t
considered projectiles.
Examples of Projectiles
•
•
•
•
•
Football
Javelin
Discus
Long jumper
Diver
A human in flight obeys the same
projectile laws as any other
object.
Factors Affecting the Trajectory
of a Projectile
• The relative height of projection
• The angle of projection (the initial angle of
the trajectory relative to horizontal)
• The speed of projection (the velocity of the
object when it is first released)
• Air resistance and wind
speed of projection
angle of projection
height of projection
Text page 336
Relative Projection Height
• This is the release height compared to the
final landing height of the projectile
Relative Projection Height
• Relative projection height = 0
Relative Projection Height
• Relative projection height = 2 m
2m
Relative Projection Height
• Relative projection height = -1.5 m
3m
1.5 m
Optimum Angle of Projection
(assuming there is no air
resistance)
• If Relative Projection Height = 0, the
optimum angle = 450
• If Relative Projection Height > 0 , the
optimum angle < 450
• If Relative Projection Height < 0 , the
optimum angle > 450
Optimum Angle of Projection
Relative Projection Height = 0
2m
Relative Projection Height = 2 m
Text page 340
3m
1.5 m
Relative Projection Height = - 1.5 m
The Components of Speed Of
Projection
The velocity at any instant in the trajectory of a projectile
can be represented as a vector that is tangent to the
trajectory.
The Components of Speed Of
Projection
By finding the vertical and horizontal components
for the instantaneous velocity vectors, you can find
the instantaneous vertical and horizontal velocities.
The Components of Speed Of Projection
EXAMPLE: A ball is thrown upward with a speed of projection
of 20 m/s. If the angle of projection is 400, calculate the
horizontal and vertical components of the speed of projection.
S = Speed of projection = 20 m/s
SH = Horizontal component
SH = (S)(cos 400) = (20 m/s)(cos 400)
SV
SH = 15.32 m/s
400
SV = Vertical component
SV = (S)(sin 400) = (20 m/s)(sin 400)
SH
SV = 12.86 m/s
Perpendicular Vectors Don’t Directly
Affect Each Other
• For example, if the projection angle of a
projectile is horizontal (the vertical
component of the projection speed is 0), it
will fall as quickly as if it is dropped with a
projection speed of 0.
Perpendicular Vectors Don’t Directly
Affect Each Other
Because the pull of gravity is unaffected by horizontal velocity, a
projectile thrown horizontally has the same vertical velocity as
an object dropped straight down. If the objects are released from
the same height they will hit the ground at the same time
(neglecting the effects of air resistance).
Acceleration Due to Gravity in Projectile
Motion
g (or ag) has the value of –9.81 m/s2 (metric
units) or –32 ft/s2 (English units) when used in
projectile motion calculations.
Because the horizontal and vertical
components of a trajectory don’t affect
each other, if air resistance is neglected
horizontal acceleration = 0 and vertical
acceleration = g (or ag).
If Relative Projection Height = 0, the
final angle and velocity of a projectile
are equal in magnitude and opposite
in direction to those of the projectile
when it is released or launched (if air
resistance is neglected).
q
q
If air resistance is neglected, the initial angle and
final angle of the trajectory are the same.
Equations of Constant
Acceleration
•
Formulas applied when acceleration is
unchanging (as in the case of the acceleration
due to gravity) Text p. 345
1) v2 = v1 + at [This is derived from the basic
formula: a = v/t = (v2 – v1)/t ]
2) d = v1t + (1/2)at2
3) v22 = v12 + 2ad
These formulas assume that:
d= displacement, v1 = initial velocity,
v2 = final velocity, a = acceleration, and t = time
PROJECTILE RANGE
Assuming an object is released and lands at the same height and there is
no air resistance: V = Initial projectile velocity VV2 = Vertical velocity at peak
VV1 = Initial vertical velocity
t = Time to reach peak
VH = Horizontal Velocity
ttotal = Total flight time = 2t
q= Angle of projection
V
VV1
VV2 = 0
aV
q
VH
Range
aV = Vertical Acceleration = -ag
t
PROJECTILE RANGE
VH = Vcos q
VV1 = Vsin q
aV = -ag = (VV2 – VV1)/t
t = (VV2 – VV1)/ -ag = (0 – VV1)/ -ag = VV1/ ag = Vsin q / ag
ttotal = 2t = 2(Vsin q / ag)
Range = (VH)(ttotal) = 2(Vcos q)(Vsin q)/ag) = 2(V2cos qsin q)/ag
V
VV1
VV2 = 0
aV
q
VH
Range
t
Air Resistance and Projectile
Motion
Air resistance (or air drag) will tend to affect
the velocity of a projectile. It tends to slow
down the horizontal component of velocity
so that the path of a projectile (if the initial
horizontal component  0) will tend to have
a steeper angle at the end than when the
projectile is launched.
Air Resistance and Projectile
Motion
Air resistance will tend to cause a projectile to fall
shorter than it would if there were no air
resistance.
Without Air Resistance
With Air Resistance