MOTION�IN TWO DIMENSIONS�(QUALITATIVE ANALYSIS)
AP Topic 1.5 Part 2a
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“Discobolus of Myron”
Myron (5th Century BC)
Learning Objective
Describe the motion of an object moving in two dimensions.
A question to get started:
At the instant a horizontally pointed cannon ball is fired, a cannonball held at the cannon’s side is released and drops to the ground. Which cannonball strikes the ground first: a) the one fired from the cannon or b) the one dropped?
What is a Projectile?
A projectile is any object that continues in motion by its own inertia and is influenced only by the downward force of gravity (and air resistance)
What is a Projectile?
Any object that continues in motion by its own inertia and is influenced only by the downward force of gravity (and air resistance)
What is a Projectile?
Any object that continues in motion by its own inertia and is influenced only by the downward force of gravity (and air resistance)
What is a Projectile?
Any object that continues in motion by its own inertia and is influenced only by the downward force of gravity (and air resistance)
Projectile Motion
Projectile motion is a special case of two-dimensional motion that has zero acceleration in one dimension (x-direction) and constant, nonzero acceleration in the second dimension (y-direction).
Projectile Motion
The parabolic path of a projectile is called its trajectory.
Examples of Projectile Motion
Projectile Motion
Some more examples of projectile motion.
Resolving Motion Into x and y Components
Motion in two dimensions can be analyzed using one-dimensional kinematic relationships if the motion is separated into components.
Side-to-side motion
(x-direction)
Resolving Motion Into x and y Components
Up-and-down motion
(y-direction)
Motion in two dimensions can be analyzed using one-dimensional kinematic relationships if the motion is separated into components.
Resolving Motion Into x and y Components
We can (and need to) consider these two motions separately when analyzing projectiles.
So which one will hit the ground first?
What force(s) act on the projectile in the y-direction while it is in projectile motion?
Only gravity acts in the y-direction (we will ignore air resistance most of the time)
So which one will hit the ground first?
What force(s) act on the projectile in the x-direction while it is in projectile motion?
There are no forces acting in the x-direction (we will ignore air resistance most of the time)
So which one will hit the ground first?
SO WHICH ONE WILL HIT THE GROUND FIRST?
ANALYZING PROJECTILE MOTION
In the y-direction, the distance between every frame (unit of time) is changing. This means there is acceleration in the y-direction.
ANALYZING PROJECTILE MOTION
Notice that in the x-direction, the distance between every frame (unit of time) is constant. This means the velocity is constant so there is no acceleration in the x-direction.
ANALYZING PROJECTILE MOTION
Let’s think about the horizontal motion of a projectile (x-direction).
What variables are involved with the motion in the x-direction?
Δx
vx
t
v0
vx
Δx
ANALYZING PROJECTILE MOTION
Now let’s think about the vertical motion of a projectile (y-direction).
What variables are involved with the motion in the y-direction?
a (due to gravity)
vy0 (y-component only)
vy (y-component only)
Δy (displacement in the y-direction only)
t
vy0
Δy
a
v
vy
v0
So to summarize what’s going on in both the x- and y-directions:
x-direction | y-direction |
Δx | Δy |
t | t |
vx | vy0 |
| vy |
| ay |
PUTTING IT ALL TOGETHER
What variable will always have the same value in both the x- and y-directions?
x-direction | y-direction |
Δx | Δy |
t | t |
vx | vy0 |
| vy |
| ay |
CONCEPTUAL QUESTION 1
Time!!! Both the x-direction movement and y-direction movement happen simultaneously for a projectile and therefore take the same amount of time.
=
a) Where is the magnitude of the vertical velocity the greatest?
CONCEPTUAL QUESTION 2
Consider the rider’s trajectory in the following picture.
AND
b) Where is the greatest horizontal velocity?
CONCEPTUAL QUESTION 2
Consider the rider’s trajectory in the following picture.
Remember: Horizontal velocity is constant.
c) Where is the vertical velocity least?
CONCEPTUAL QUESTION 2
Consider the rider’s trajectory in the following picture.
Remember: At the top of the path, the bike is not moving up or down. It is transitioning from upward motion to downward motion.
d) What type of path is the rider following?
CONCEPTUAL QUESTION 2
Consider the rider’s trajectory in the following picture.
Remember: All projectiles (except in freefall) follow a parabolic path.
Ray Gunn is playing with a radio-controlled race car on a balcony of a 6th floor condominium. An accidental turn sends the car through the railing and off the edge of the balcony.
b) Does the horizontal distance the car falls depend on the velocity with which it speeds off the edge?
YES!
CONCEPTUAL QUESTION 3
= slow vi
= medium vi
= fast vi
Δxslow
Δxmed
Δxfast
Δy is the same for all three vi values
Warren Peace is flying an airplane at constant velocity and constant altitude. He drops a bomb from the bottom of the plane. Ignoring air resistance, where will the bomb be relative to the plane when the bomb hits the ground?
CONCEPTUAL QUESTION 4
CONCEPTUAL QUESTION 4
It will be directly beneath the plane since it will continue to travel in the x-direction with the same velocity that it was dropped with.
Jim Shorts hits a golf ball. What is the acceleration of the ball (ignoring air resistance):
It is the same at all points along the path: 9.8 m/s2 downward. There is no acceleration in the x-direction.
CONCEPTUAL QUESTION 5
KEY CONCEPTS FOR PROJECTILE MOTION