Projectile Motion
Physics level 1 – Mr. Creery
y vs. x plane
y vs. t plane
x vs. t plane
Projectile Motion Terms
• X velocity component – The velocity of just
the x axis found by cos(theta)*v
• Y velocity component - The velocity of just the
y axis found by sin(theta)*v
• Range - How far a projectile moves along the
x-axis.
• Height – How far a projectile moves along the
y-axis
• Time – Independent variable which is the
same on both the x and the y axis.
Projectile Motion Concepts
• Vertical (y) and Horizontal (x) components are
independent of each other.
• There is no acceleration on the horizontal (x)
component.
• The acceleration due to gravity is -9.8m/s2 on the
vertical (y) axis.
• The vertical time is the same as the horizontal
time.
• Velocity at the top of the trajectory is 0m/s and
the acceleration is always the same.
Projectile Motion Charts
Chart shows the horizontal and vertical
velocity vectors.
Chart show the vertical and horizontal
constants.
A projectile question chart.
Vertical and Horizontal Motion
y vs. x plane
y vs. t plane
x vs. t plane
Projectiles Launched at an Angle
Projectiles Launched at an Angle
5s
4s
3s
2s
1s
5 m 20m 45m 80m 125m
• This diagram shows that there
are specific vertical distances for
the cannonball from the
position of where it should be
(with no gravity) to where it
actually is (with gravity).
• These distances are the SAME
distances it would fall if it were
dropped and were undergoing
free fall for the given amount of
time.
Projectile Motion Vertical Graphs
Graphs for an object thrown upwards
with an initial velocity vi . The object
takes time to reach its maximum
height after which it falls back to the
ground. (a) position vs. time graph (b)
velocity vs. time graph (c) acceleration
vs. time graph.
Projectile Motion Horizontal Graphs
Projectile Motion Examples
• Trajectories
• For objects launched at an angle, you have to do a little more work
to determine the initial velocity in both the horizontal and vertical
directions.
• For example, if a football is kicked with an initial velocity of 40
m/s at an angle of 30° above the horizontal.
• First, break the initial velocity vector up into x- and ycomponents.
• Then, use the components for initial velocities in your horizontal
and vertical to solve for the time on the y and the range on the x.
• Don’t forget that symmetry of motion also applies to the parabola
of projectile motion. For objects launched and landing at the same
height, the launch angle is equal to the landing angle. The launch
velocity is equal to the landing velocity. And if you want an object
to travel the maximum possible horizontal distance (or range),
launch it at an angle of 45°.
Height
• The vertical distance a
projectile falls below an
imaginary straight-line
path increases
continually with time
and is equal to h=1/2at2
or 4.9t2 meters.
Height
• The diagram above shows the path of a projectile with velocity
vectors drawn in for different points on its parabolic path. The
vectors have been resolved into their x and y components.
Height
• The horizontal component is always the same – the ball
moves equal distances in equal time intervals.
• Note that ONLY the vertical component changes – it decreases
going upward against gravity and increases going downward
with gravity. The projectile accelerates due to Earth’s gravity.
• Is the velocity of the projectile at its highest point zero?
Range
• Now suppose 2 projectiles are
launched at the same speed but
different angles. The projectile
launched at the greater angle will
have a velocity vector with a greater
vertical component and a smaller
horizontal component.
• The greater vertical component
results in a higher path; the smaller
horizontal component results in less
range (distance).
Example Question: Herman the human cannonball is
launched from level ground at an angle of 30° above
the horizontal with an initial velocity of 26 m/s. How
far does Herman travel horizontally before reuniting
with the ground?
• Answer: Our first step in solving this type of problem is
to determine Herman's initial horizontal and vertical
velocity. We do this by breaking up his initial velocity
into vertical and horizontal components:
• vi(x) = vi * cos() = 26(m/s) * cos(30o) = 22.5m/s
• vi(y) = vi * sin() = 26(m/s) * sin(30o) = 13m/s
• Next, we'll analyze Herman's vertical motion to
find out how long he is in the air. We'll analyze his
motion on the way up, find the time, and double
that to find his total time in the air:
List Variables
•vi=13 m/s
•vf=0
•d=?
•a=-9.81 m/s2
•t=?
Solving For t
t1/2= vfy-viy/-a
t1/2= 0-13.0m/s/-9.8m/s*s = 1.3265sec
t= 1.33*2 = 2.65sec
• Now that we know Herman was in the air 2.65s, we can
find how far he moved horizontally, using his initial
horizontal velocity of 22.5 m/s.
• •vi=22.5 m/s
•vf=22.5 m/s
•d=?
•a=0
•t=2.65s
Answer
vx=d/t so d(x) = vx * t or d = (22.5m/s*2.65s)=59.6m
• Therefore, Herman must have traveled 59.6m horizontally
before returning to the Earth.
Question
For projectile shot at an angle of 60 degrees.
What component is being depicted in the following graphs and why?
Answer
• Each of these graphs show acceleration. This
would have to be the y-axis because there is
no acceleration on the x-axis.
Question
• You should have found that launch angle is a
key variable.
– What angle causes the largest height?
– What angle causes the largest range?
Answers
• 90 degrees to the ground causes the greatest
height.
• 45 degrees causes the greatest range.
Practice Problems
1.) A cannon ball is fired at an angle of 36
degrees with an initial velocity of 20m/s. What
are the vertical and horizontal components?
2) A cannon ball is fired at an angle of 36 degrees with an initial velocity of 20m/s.
How long is the ball in the air?
3) A cannon ball is fired at an angle of 36 degrees with an initial velocity of 20m/s.
How far does the ball travel?
Answers
1.) vi(x) = vi * cos(theta) = 20(m/s) * cos(36o) = 11.8m/s
vi(y) = vi * sin(theta) = 20(m/s) * sin(36o) = 16.2m/s
2) t1/2= 0-16.2m/s/-9.8m/s*s = 1.65sec * 2 = 3.3sec
3) dx= 3.3sec * 11.8m/s = 39.0m
Learning Projectile Motion
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Learning Objectives
Your Learning Goal is to be able to:
● Predict how varying conditions will effect the path of a projectile and explain
your reasoning.
● Use the PhET simulation to confirm or disprove your predictions.
● Explain why your predictions were correct or incorrect.
Your Grade! - Click here for the worksheet that you will turn in for credit.
Projectile_Motion_WS
Click Here for steps to solving Projectile Motion
Problems. Projectile_Motion_Steps
Click Here to see vector notes and homework answers.
Vector_Notes_Homework
References:
http://aplusphysics.com/courses/regents/kinematics/regents_projectile_motion.h
tml
http://phet.colorado.edu/en/simulation/projectile-motion