http://www.ck12.org Chapter 3. Two-Dimensional MotionInertial FramesThere are endless examples of relative motion. Suppose that you’re in an elevator that is rising with a constant speed
of 2 m/s relative to the ground. If you release a ball while in this reference frame, how will the motion of the ball
differ than had you dropped the ball while standing on the ground?
Had you been asleep in this reference frame and woke after the compartment was in motion, you would have no idea
you were in motion. There is no experiment that can be performed to detect constant velocity motion. If you’ve ever
traveled in a jet moving 1000 km/h (about 600 mph) with no air turbulence, then you know from firsthand experience
that you felt motionless. After the brief acceleration period, you can no longer sense the motion of the elevator. The
ball will move as if it has been released in the reference frame of Earth.
As a general statement, we consider all constant velocity reference frames to be equivalent. This idea is known as
The Galilean Principle of Relativity. Constant-velocity reference frames are calledinertial framesof reference.
An “at-rest” reference frame is an arbitrary construct. If you’re traveling with a constant velocity in your car, the
reference frame of the car is an at-rest frame. The elevator compartment moving 2 m/s is an at-rest frame with
respect to the compartment. You- who are sitting and reading this firmly placed on the earth- consider yourself
to be in an at-rest reference frame. But you know the Earth itself is in motion. It rotates about its axis with a
speed of about 1600 km/h (1000 mph) at the equator, and it orbits the sun with an average speed of 108,000 km/h
(67,000 mph). In fact, the Earth isn’t even an inertial frame of reference, since it rotates about its axis and its orbital
speed varies. (Remember, velocity is constant only if its magnitude and direction do not change—objects in circular
motion do not qualify!) We usually approximate the Earth as an inertial frame of reference since we do not readily
sense the earth’s acceleration. Objects on Earth’s surface have a maximum acceleration due to its rotation of about
0 .03 m/s^2 —which we don’t typically concern ourselves with since the acceleration due to gravity is 10 m/s^2.Relative Motion: Part 1TheFigure3.13, a moving walkway, provides us with our first example of relative motion. Let us consider two
Cartesian coordinate systems. One is attached to the “stationary” Earth. The other is attached to a walkway moving
with a constant horizontal velocity of 1 m/s with respect to the earth. If a ball is thrown with an initial horizontal
velocity of 3 m/s in the direction the walkway is moving by a person standing on the walkway, what horizontal
velocity does a person standing on the ground measure for the ball? The person in the Earth frame sees the ball
having a combined velocity of 4 m/s. The person in the “moving frame” will measure it as 3 m/s. According to The
Principle of Galilean Relativity, the velocity,V, seen from the at-rest frame is additive, that is,V= 1 m/s+ 3 m/s.
SeeFigure3.14.
http://demonstrations.wolfram.com/RelativeMotionInASubwayStation/
Two cars are headed toward each other. CarAmoves with a velocity of 30 mph due east and CarBwith a velocity of
60 mph due west, relative to “at-rest” earth. SeeFigure3.15.
a. What is the velocity of carBrelative to the velocity of carA?
We define motion to the east as positive (+30 mph), and motion to the west as negative (-60 mph).
From our previous statements regardingrelative velocitywe can “feel” that the relative velocity is greater than either
speed: If we define the relative velocity (the velocity of carBrelative to the velocity of carA) as:~Vba=~Vb−~Va,
then− 60 − 30 =− 90 m ph, a person in carAsees carBmoving west at 90 mph. The person in carAsees himself as
“motionless” while carBis moving toward him with carB’s speed and his own speed which he does not perceive.
b. What is the velocity of carArelative to the velocity of carB? This means we are assuming carBis our “at rest”
coordinate system.~Vab=~Va−~Vb=30 mph−(−60 mph) = +90 mph. A person in carBsees carAmoving east at 90 mph.
The person in carBsees himself as motionless, while carAis moving toward him with carA’s speed and his own
speed, which he does not perceive.