Projectile Motion
Projectiles at an Angle
Projectiles at an Angle
• Last lecture, we
discussed projectiles
launched horizontally.
• Horizontal projectiles
are just one type of
projectile problem that
we can discuss.
• They are also one of
the easiest
Projectiles at an Angle
• Now we will discuss
projectiles that are
launched at an angle
other than 0°.
• They still share many
of the same
characteristics of
horizontal projectiles
Projectiles at an Angle
• As the animation to the
right shows, horizontal and
vertical movement are still
independent of each other.
• If we neglect air resistance,
any projectile launched
from a moving object will
land in the same place it
was launched if the moving
object maintains the same
speed.
Projectiles at an Angle
• Lets analyze the
following problem and
attempt to solve it.
• A football is kicked with
a velocity of 25 m/s at
an angle of 60°.
Calculate the range,
total time in the air, and
maximum height of the
football’s path.
Solving Problems
• We need to start just like we did with horizontal
projectiles and make an x and a y column.
• We need to determine what part of the velocity of
the ball is in the horizontal and what part is in the
vertical.
• We do this by finding the x and y components of
the 25 m/s velocity at the angle of 60°.
• By using trigonometry, we find the following:
• Vx = Cos60(25m/s) =12.50 m/s
• Vy = Sin60(25m/s) = 21.65 m/s
Solving Problems
• X – column
• Y – column
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Vi = 12.50 m/s
t = 4.42 sec
d = vt
d = (12.50)(4.42)
d = 55.25m
Vi = 21.65 m/s
a = -9.8 m/s
Vf = 0 (at max height)
Vf = Vo + at
0 = 21.65 + (-9.8)(t)
t = -21.65/-9.8
t = 2.21 seconds
Vf2 = Vi2 + 2ad
0 = (21.652) + (2)(-9.8)(d)
d = -468.72/-19.6
d = 23.91 m
Important Considerations
• One important consideration is that we must
make the acceleration of gravity NEGATIVE
in these calculations.
• The reason has to do with the fact that we
have an object moving both up and down.
• We must use up as a positive value and
down as a negative value.
Important Considerations
• Another assumption we must make is that at
the maximum height, Vf in the y direction will
be 0.
• Understand that at the exact instant that the
ball reaches the maximum height, it has no
velocity up or down in the vertical direction.
Important Considerations
• Notice that when we
calculate time in the y
column, we are only
finding how long it
takes to reach the
maximum height.
• Time must be
multiplied by 2 to
determine the length of
the entire trip