Frame of Reference
It depends on your point of view
First, a simple example
Nathan Eovaldi can throw a baseball 100 �mph. If he throws a fastball on an airplane �which is flying 500 mph, how fast does the �baseball go?
It depends!
Easy one first
To someone sitting on the plane, it’s simple: The baseball has a speed of 100 mph. If it’s thrown to the back of the plane, and we consider that direction “negative,” then the velocity is -100 mph.
To anyone sitting on the plane, it appears as if the plane is holding still (vplane = 0 mph), and any other velocity must be compared to the plane. The ground appears to be moving -500 mph!
Why did anyone ever think this was correct?
A little harder
To an observer outside the airplane, we have to account for both the velocity of the airplane and the velocity of the baseball, each of which have their own velocity vector.
That’s right, we’re adding vectors. In one dimension, thankfully. (for now)
1D Vectors
If the baseball is thrown toward the front of the plane (let’s call that direction “positive”), the velocity vectors stack:
500 + 100 = 600 mph
If thrown toward the rear of the plane, the velocities are still simply added, but they partially cancel out:
500 + -100 = 400 mph
An observer on the ground would still see the baseball moving forward, even though it’s thrown backward.
So how fast was it “really” moving?
It depends!
There’s no such thing as a “correct” universal frame of reference.
While it may seem like the baseball was “really” moving 600 mph, that implies Eovaldi can throw a baseball 600 mph. I assure you he cannot.
What if we observed the scenario from the Moon? It’d be moving thousands of mph!
Without knowing the frame of reference, we can’t answer the question!
Another example
An object might be “really” moving �forward, but appear to have a negative �velocity, given a specific frame of reference.
This happens all the time when you pass a car on the highway. To you, the slower car appears to be moving backward relative to your car, with a negative velocity, even though it’s obviously not driving in reverse.
In this picture, which car is passing the other? Can we tell?
Math from a moving perspective
If objects A and B each have their own velocities vA and vB, with respect to a specific frame of reference (like the ground):
From any object’s perspective, subtract its own “true” velocity from the “true” velocity of the observed object.
from Object A’s perspective from Object B’s perspective� vB = vB - vA vA = vA - vB
Practice 1
Agent Landauer tracks down Dr. Imabadguy and catches him �trying to escape to Pottsylvania on the �Orient Express, which moves 30 m/s.
Imabadguy runs toward the rear of the train �at -4 m/s, while Agent Landauer pursues him at -5 m/s. From Agent Landauer’s perspective, at what speed is Dr. Imabadguy getting closer to him?
-4 - -5 = +1 m/s
Practice 2
Imabadguy reaches the end of the train and jumps -4 m/s off the back of the train, which is still moving 30 m/s.
With what horizontal velocity does Imabadguy hit the ground?
-4 + 30 = 26 m/s
Practice 3
Dr. Imabadguy survives the landing, then while on the ground, he fires two shots at Agent Landauer, who is still on the train, and is bulletproof. The bullets leave Imabadguy’s gun at a velocity of 400 m/s. With what velocity do the bullets impact Agent Landauer?
400 - 30 = 370 m/s
Airplane baseball again, in 2D!
What if Eovaldi threw the baseball to the side of the plane, let’s say to the right? We’d have to combine perpendicular vectors. That’s right, trig!
To find the magnitude, Pythagorean Theorem:� √(5002 + 1002) = 509.902 mph
To find the direction, inverse trig:� tan-1(100/500) = 11.310° right of forward
Full answer: 509.902 mph, 11.310° right of forward
Relative velocity and position are two different things
Two objects begin from rest at the same position, then have these positions:
x1(t) = 1.5t blue� x2(t) = t2/4 red
At what time does object 2 appear to have a velocity of 0, as observed from object 1?
Their velocities must be equal.
Find velocities and set them equal
Velocity is the derivative of position, so:� v1(t) = 1.5� v2(t) = t/2
Set them equal to each other:� 1.5 = t/2� t = 3 s�You should be able to (kinda) tell this is where the lines have the same slope.
t = 6 s is the time when they have the same position, and object 2 passes object 1.
How to “show work” on calculator problems
On free response questions, you’re almost always required to show work or write a justification for your answer.
How do you “show work” when you did the whole thing in a calculator?
If you used your calculator to find:�∫05 t3/2 + 9t - 2 dt = 124.861 m
WRITE THAT WHOLE LINE, not simply “124.861 mm”
In other words, write down what you put in the calculator!
Five Minutes
What are the most important factors for choosing a job?
What would make you quit a job?
Trajectories
x and y are independent
Trajectory
A trajectory is the path a projectile makes while in free-fall.
In an Earth-like system with �constant acceleration (often �due to gravity), a trajectory �will take the shape of a �parabola.
Dimensional Independence
x and y function entirely independent of each other.�If you were to throw a baseball horizontally or drop one from the same height, they’d hit the ground at the same time!
Because trajectories usually don’t have any acceleration in the x direction, vx is usually constant.�vy will typically be affected by gravity (ay = g) and will therefore change.
Trigonometry
Remember the Unit Circle?�cosine goes with x�sine goes with y
It’s also helpful to know soh-cah-toa.
Do this in order to break down a vector into x and y components:
v0x = v0·cos(θ)�v0y = v0·sin(θ)�θ is the launch angle above the horizontal.
Fun Facts
Example (pt 1)
Mr. Landauer decides the trash can is too good for his piece-of-defecation inner tube which keeps getting flats. He drop kicks it with an initial height of 0.213 m and an initial velocity of 37 m/s, 42° above the horizontal. It winds up landing on top of a parked car with a height of 1.861 m. How far away does it land?
What do we know?
What are we looking for?
Is there an equation that relates these variables?
Is there any other variable we’ll need to solve for?
Example (pt 2)
What do we know?
y0 = 0.213 m� yf = 1.861 m� v0 = 37 m/s� 𝛳 = 42° above the horizontal� ax = 0 m/s2� ay = -9.8 m/s2
What are we looking for?� 𝛥x = ? m
Is there an equation(s) that relates these variables?� xf = x0 + v0xt + (axt2)/2� yf = y0 + v0yt + (ayt2)/2
Is/are there any other variable(s) we’ll need to solve for?
v0x = ? m/s� v0y = ? m/s� t = ? s
Example (pt 3)
Let’s start by finding vx and vy.
v0x = 37·cos(42°) = 27.496 m/s�v0y = 37·sin(42°) = 24.758 m/s
xf = (axt2)/2 + v0xt + x0 has two unknown variables (xf and t), so it’s not useful yet.
yf = (ayt2)/2 + v0yt + y0 has only one unknown!
Handy Rule: To solve any system of equations, you need at least as many equations as you have unknown variables
1.861 = -4.9t2 + 24.758t + 0.213
This is a quadratic equation. The only two good ways to solve it are with the quadratic formula or by graphing.
You’d need a calculator anyway. Graph it!
y = 1.861
y = -4.9x2 + 24.758x + 0.213
See where they intersect.
Example (pt 4)
We got a problem: they intersect more than once!��Think though: Which one of these �answers makes more sense in the �context of the problem?
t = 4.986 s
Now we can use this to find xf.
Example (pt 5)
xf = (axt2)/2 + v0xt + x0
ax = 0 m/s2�v0x = 27.496 m/s
x0 = 0 m�t = 4.986 s�xf is what we’ve been looking for all along!
�xf = 0 + (27.496 · 4.986) + 0 = 137.096 m
Does this answer make sense? It’s a little more than a football field.
#7 from Assignment 1.5
A projectile is launched from level ground at an initial velocity v0 and angle θ as shown in the figure. At the instant the projectile is moving horizontally, it hits a wall and then bounces directly back at half the speed it had just before impact. Assuming the projectile is in contact with the wall for a negligible time, find an expression using v0, θ, and g, for the time the projectile is in the air from hitting the wall to reaching ground level again.
We want t, we have to express it in terms of v0, θ, and fundamental constants.
#7 from Assignment 1.5
Time depends only on the y dimension. Horizontal movement has no effect on how fast things fall.
v0y = v0sin(θ) this is the only v we’ll use!�g = a
Is there a kinematics formula that relates v, a, t, and nothing else?
v = v0 + at
#7 from Assignment 1.5
v = v0 + at
When the ball hits the ground, v = -v0, so:�-v0 = v0 + at�
Now we simply sub in v0sin(θ) and g and solve for t:
-v0sin(θ) = v0sin(θ) + gt�-2v0sin(θ) = gt
t = v0sin(θ)/g since g is negative, it cancels out the existing negative
Before Next Time
Our first test!
and some review first
next time