stationary here on Earth, you must also conclude that the entire universe revolves
around the Earth (a dubious conclusion, though a common one for centuries).
Reference frames define your perception and measurements of motion. If you are in a
car moving at 80 km/h, another car moving alongside you with the same velocity will
appear to you as if it is not moving at all. As you drive along, objects that you ordinarily
think of as stationary, such as trees, seem to move rapidly past you. On the other hand,
someone sitting in one of the trees would say the tree is stationary and you are the one
moving by.
A reference frame is more than just a viewpoint: It is a coordinate system used to make
measurements. For instance, you establish and use a reference frame when you do lab
exercises. Consider making a series of measurements of how long it takes a ball to roll
down a plane. You might say the ball’s starting point is the top of the ramp. Its x position
there is 0.0 meters. You might define the surface of the table as having a y position of
0.0 meters, and the ball’s initial y position is its height above the table. Typically you
consider the plane and table to be stationary, and the ball to be moving.
Two reference frames are shown on the right. One is defined by Joan, the woman
standing at a train station. As the illustration in Concept 1 shows, from Joan’s perspective, she is stationary and the train is moving to the right
at a constant velocity.
Another reference frame is defined by the perspective of Ted who is inside the train, and considers the train stationary. This reference frame is
illustrated in Concept 2. Ted in the train perceives himself as stationary, and would see Joan moving backward at a constant velocity. He would
assign Joan the velocity vector shown in the diagram.
It is important to note there is no correct reference frame; Joan cannot say her reference frame is better than the reference frame used by Ted.
Measurements of velocity and other values made by either observer are equally valid.
Reference frames are often chosen for the sake of convenience (choosing the Earth’s surface, not the surface of Jupiter, is a logical choice for
your lab exercises). Once you choose a reference frame, you must use it consistently, making all your measurements using that reference
frame’s coordinate system. You cannot measure a ball’s initial position using the Earth’s surface as a reference frame, and its final position
using the surface of Jupiter, and still easily apply the physics you are learning.
Reference frames
Measurements of motion defined by
reference frame4.15 - Relative velocity
Observers in reference frames moving past one another may measure different
velocities for the same object. This concept is called relative velocity.
In the illustrations to the right, two observers are measuring the velocity of a soccer ball,
but from different vantage points: The man is standing on a moving train, while the
woman is standing on the ground. The man and the woman will measure different
velocities for the soccer ball.
Let’s discuss this scenario in more depth. Fred is standing on a train car and kicks a ball
to the right. The train is moving along the track at a constant velocity. The train is Fred’s
reference frame, and, to him, it is stationary. In Fred’s frame of reference, his kick
causes the ball to move at a constant velocity of positive 10 m/s.
The train is passing Sarah, who is standing on the ground. Her reference frame is the
ground. From her perspective, the train with the man on it moves by at a constant
velocity of positive 5 m/s.
What velocity would Sarah measure for the soccer ball in her reference frame? She
adds the velocity vector of the train, 5 m/s, to the velocity vector of the ball as measured
on the train, 10 m/s. The sum is positive 15 m/s, pointing along the horizontal axis.
Summing the velocities determines the velocity as measured by Sarah.
Note that there are two different answers for the velocity of one ball. Each answer is
correct in the reference frame of that observer. For someone standing on the ground,
the ball moves at 15 m/s, and for someone standing on the train, it moves at 10 m/s.
The equation in Equation 1 shows how to relate the velocity of an object in one
reference frame to the velocity of an object in another frame. The variable vOA is the
velocity vector of the object as measured in reference frame A (which in this diagram is
the ground). The variable vOB is the velocity of an object as measured in reference
frame B (which in this diagram is the train). Finally, the variable vBA is the velocity of
frame B (the train) relative to frame A (the ground). vOA is the vector sum of vOB and
vBA.
An important caveat is that this equation can be used to solve relative velocity problems
only when the frames are moving at constant velocity relative to one another. If one or
both frames are accelerating, the equation does not apply.
We use this equation in the example problem on the right. Now Sarah sees the train moving in the opposite direction, at negative 5 m/s. It is
negative because to Sarah, the train is moving to the left along the x axis. Fred is on the train, again kicking the ball from left to right as before.
Here he kicks the ball at +5 m/s, as measured in his reference frame. To Sarah, how fast and in what direction is the ball moving now?
Observer on train
Measures ball velocity relative to train
Observer on ground
Train is moving
Velocity = sum of ball, train velocities