3. Topic objectives
State the independence of the vertical and the horizontal
components of velocity for a projectile in a uniform field.
Describe and sketch the trajectory of projectile motion as
parabolic in the absence of air resistance.
Describe qualitatively the effect of air resistance on the
trajectory of a projectile.
Solve problems on projectile motion.
Remarks:
1. Proof of the parabolic nature of the trajectory is not required.
2. Problems may involve projectiles launched horizontally or at any angle above or below the
horizontal. Applying conservation of energy may provide a simpler solution to some problems
than using projectile motion kinematics equations.
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4. What is a projectile?
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When a body is in free motion, (moving
through the air without any forces apart
from gravity and air resistance), it is called a
projectile
Normally air resistance is ignored so the only
force acting on the object is the force due to
gravity
This is a uniform force acting downwards
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5. Types of projectiles
There are three types of projectile depending on
the value of the angle between the initial velocity
and the x-axis.
1. θ = 0
horizontal projectile
2. θ = 90 vertical projectile (studied earlier)
3. θ = θ which is the general case.
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6. HORIZONTAL PROJECTILES
•Horizontal Projectiles are easiest to work
with
•only formula used in horizontal (x)
direction is:
x = ut
Remark: in case of horizontal
projectiles select the direction
of the y-axis to be downward.
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8. HORIZONTAL PROJECTILES
•vertical (y) direction is just freefall
•all of the initial velocity is in the x
direction
•So,
u
t1
t2
t3
t4
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9. EXAMPLE
A person decides to fire a rifle horizontally at a
bull’s-eye. The speed of the bullet as it leaves the
barrel of the gun is 890 m.s-1. He’s new to the
ideas of projectile motion so doesn’t aim high and
the bullet strikes the target 1.7 cm below the
center of the bull’s-eye.
What is the horizontal distance between the rifle
and the bull’s-eye?
start by drawing a picture:
label the explicit givens
890 m.s
1
1.7 cm
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10. EXAMPLE
What is the horizontal distance between the rifle
and the bull’s-eye?
X
ux
890 m.s
Y
1
y
ay
uy
1.7 cm
m
9.8 2
s 1
0 m.s
890 m.s
1m
100cm
0.017m
1
1.7 cm
want:
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dx
horizontal distance
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11. EXAMPLE
which equation do we use?
use
y
0
1 2
ayt uyt
2
to find time
rewrite equation for t
t
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2y
ay
2(0.017)
9.8
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0.059 s
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12. EXAMPLE
Use t and ux to solve for x
x
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uxt
(890)(0.059)
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52.4 m
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14. NON-HORIZONTAL PROJECTILES
• vx = ux is still constant
• uy is also constant
•only difference with non-horizontal is
that vy is a function of time
u
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18. VISUALIZING PROJECTILES
•first enter vectors
•focus on vx
•focus on vy
no vy at the top!
vy decreases as it
rises!
by how much per second?
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24. VARIED ANGLES
•which projectile angle shoots highest?
•larger θ means faster uy
•which projectile angle shoots farthest?
•45° has perfect balance of fast vx and long flight
time.
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25. If projectiles are launched at the same speed, but at different
angles, the height and range is of the projectile are affected.
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26.
27. Solving Problems Involving Projectile
Motion
1. Read the problem carefully, and choose the
object(s) you are going to analyze.
2. Draw a diagram.
3. Choose an origin and a coordinate system.
4. Decide on the time interval; this is the same in
both directions, and includes only the time the
object is moving with constant acceleration g.
5.Examine the x and y motions separately.
28. Solving Problems Involving Projectile
Motion (cont.)
6. List known and unknown quantities.
Remember that vx never changes, and that
vy = 0 at the highest point.
7. Plan how you will proceed. Use the
appropriate equations; you may have to
combine some of them.
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29. When the effect of air resistance is significant,
the range of a projectile is diminished and the
path is not a true parabola.
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30. In the case of air resistance, the path of a high-speed
projectile falls below the idealized path and follows
the solid curve.
Computer-generated trajectories of a baseball with and
without drag.
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