Have you ever noticed when you're speeding down the highway at \(65\,\mathrm{mph}\) that the cars next to you look like they are moving slowly? But when you get out of the car and stand on the side of the road, the vehicles nearly knock you off your feet: they're moving extremely fast. How does that make any sense? The other cars seem to be slow because you are in a moving reference frame relative to them. Since your frame of reference is moving along the road with you, you see the motion of other vehicles relative to your own. This article will define what a frame of reference is and explain the characteristics of reference frames through relevant examples.
Frame of Reference Definition
Yes, that's "reference" up there, not "reverence." Don't worry, there aren't some frames that we have to revere above others; physics does not have a social hierarchy—this is not a history article, thank goodness.
A frame of reference is a coordinate system with objects and characteristics that we define to tackle a particular physics problem.
For example, when we talk about falling objects, we usually imply that upwards is positive, and downwards is negative. Defining which direction is negative is the frame of reference we operate inside to solve our gravity problem. All throughout your physics experience, you have been setting frames of reference to allow you to answer questions.
Frame of Reference Characteristics
The great thing about a frame of reference is that you can define all the characteristics! Don't get me wrong; you still have to obey all the laws of physics. You can't just say, "I'm going to define Newton's Second Law to be invalid!" You can, however, define the characteristics of your system to benefit you the most and make the most sense for you. For example, maybe you dropped an object above a fan blowing upward, causing the object to rise into the air. Seeing how the object is moving upward, it might not be such a bad idea to define the upward direction as negative to see how much the force of the fan is counteracting gravity!
For example, in Fig. 2 below, we choose to define the up direction as positive. The free-body diagram off to the left shows that the force of the fan is much greater than the gravity of the ball. By defining upwards as positive, we can subtract \(F_g \) from \(F_\text{fan}\) and figure out how much the fan counteracts gravity.
Fig. 1 - In this problem, we define up in the \(y\) direction as positive
The Observer
We cannot talk about reference frames without talking about observers. Physics is all about how objects in the universe interact with each other. However, depending on location, one person might see an interaction differently than another.
For example, a person standing on the ground may look in the sky and see a plane. Then, two minutes later, that person looks up again and sees the same plane in the sky. "It didn't travel that far," he may think to himself because he can still see it. People in the plane, however, would think differently. In those short two minutes, they would pass the airport, maybe some mountains, and probably an entire city. The location from which an interaction is observed can mean everything when describing how it all happened.
Perspective is key. Similar to how people can get into arguments because they see things differently, if a proper frame of reference is not defined, physicists will see a problem differently and come up with diverse answers. Therefore, by choosing a reference frame, one is also choosing to define the direction and magnitude of the parts of the physical system measured by an observer in that frame of reference.
Inertial Reference Frames
Recognize that measurements within a given reference frame may be converted to measurements within another reference frame.
Therefore, using formulas and calculations, different reference frames can be compared and related to each other. Think of unit conversions: to convert from meters to kilometers, you divide by \(1000\). Converting measurements in one frame of reference to another is no different. However, the conversion process is often more complex.
There are two types of reference frames where these conversions can take place: inertial and non-inertial.
Inertia is the innate characteristic of an object to resist a change in motion.
Inertial Reference Frames
Inertial reference frames assume that acceleration is constant and Newton's first law is directly applicable. Remember that Newton's first law is that an object in motion will remain in motion unless acted upon by an outside force: sometimes, we refer to this fundamental physics principle as the law of inertia because it has everything to do with an object's resistance to a change in motion.
So, what does it mean for Newton's first law to be directly applicable? The first phrase, "an object in motion will remain in motion," identifies an inertial reference frame. Any frame of reference with no external net force acting on it is an inertial reference frame because its acceleration will remain constant relative to all other frames.
The acceleration of any object is the same as measured from all inertial reference frames.
A reference frame involving one stationary object is an excellent example of an inertial reference frame. For example, a girl standing on the sidewalk next to the street watching cars whiz past her is in her own inertial frame of reference. She is not moving, and no forces are acting on her to make her move. Therefore, she will see motion around her as independent in its own sphere, without any variation due to her own movement. If she were moving as well, for instance, the motion of cars around her would be harder to track because we would have to add her motion to the motion of the vehicles.
Non-inertial Reference Frames
Non-inertial reference frames address the second part of Newton's first law: "unless acted upon by an outside force." These frames of reference have an acceleration with respect to an inertial frame - or in other words, an accelerator at rest in a non-inertial frame of reference would detect a non-zero acceleration. This results in classical mechanics explaining the motion of bodies using fictitious forces (also known as pseudo-forces), such as centrifugal force.
A typical example of non-inertial reference frames is a frame that involves rotational motion. An object travelling along a circular path has an acceleration; therefore, a reference frame centered on the object must also have an acceleration, and would therefore be non-inertial. If you imagine a free-body diagram of the moon, there is only one real force acting on it - the pull of gravity. In order for the moon to be in equilibrium and not accelerate out of our frame of reference, the fictitious centrifugal force balances the gravitational force.
Fig. 3 - The Moon constantly accelerates as it orbits the Earth, making a viewpoint centered on the moon a non-inertial frame of reference.
Frame of Reference and Motion
Frames of reference help us to understand motion better. Although every physicist would like it to be, the real world is not ideal. In a world designed for simple analysis of motion, every object and person would take its turn to move. Can you imagine?
Okay, car number 364, it is your turn to move forward 3 meters. No, not 354, 364. There you go...365, where is 365?
The reality is that everything is moving relative to each other. Frames of reference allow us to "frame" motion with respect to other things that are moving.
I know, pretty confusing, right? There's a reason that physicists did not unlock how relative motion and the speed of light get along until Albert Einstein. Remember how we talked about converting the measuring of quantities from one reference frame to another? Let's say you were moving with some velocity in a car. To calculate your velocity relative to another car's velocity, you simply add the observed car's velocity to your own.
Therefore, an observed object's velocity is found by adding the observed object's velocity and the observer's velocity. This is done through the addition or subtraction of vectors. This addition or subtraction looks something like this:
$${\vec v_a}_b= ({\vec v_a}_d+ {\vec v_d}_b).$$
To translate this formula into laymen's terms, it means, "The velocity of object \(a\) in \(b\)'s frame of reference is equal to the velocity of \(a\) in \(d\)'s frame of reference plus the velocity of \(d\) in \(b\)'s frame of reference."
Note that the acceleration of an object is the same for all observers in all inertial reference frames. In the AP Mechanics test, all frames of reference can be assumed to be inertial unless otherwise stated.
Frame of Reference Examples
Now that our head is spinning, let's try to straighten it out by actually applying this to the real world.
Example 1
First, we'll begin with adding relative velocities.
On the image above, you see a spaceship traveling toward earth at a velocity of \(0.50\) times the speed of light. A canister is then ejected towards the Earth at \(0.75\) times the speed of light, as measured by an observer in the spaceship. The speed of light is
$$c=3*10^8\,\mathrm{\frac{m}{s}\\}\mathrm{.}$$
What is the velocity of the canister as seen by an observer from Earth?
First, write down everything that we know in a table.
Object | Formula Equivalent |
Canister | \(a\) |
Earth | \(b\) |
Spaceship | \(d\) |
Now add the velocities together to get how an observer from earth would see it using
$${{\vec v}_a}_b=({{\vec v}_a}_d+{{\vec v}_d}_b).$$
Initially, the canister and the spaceship were moving together, then when it gets ejected, the canister is moving \(0.75\) times the speed of light faster than the spaceship. Therefore, the velocity of the canister with respect to the spaceship is \(0.75\) times the speed of light. Written mathematically, this would be:
$${{\vec v}_a}_d = 0.75c\,.$$
The spaceship is already moving at \(0.50\) times the speed of light, therefore, its velocity relative to an observer from the earth is \(0.50\) times the speed of light. Therefore, an observer would see it as having a velocity of
$${{\vec v}_d}_b = 0.50c\,.$$
We then add our velocities together to figure out how an observer would perceive the canister's velocity from the earth. Using our relative velocities equation gives
$${{\vec v}_a}_b = 0.75c+0.50c\,.$$
Then we add our two velocity vectors,
$$0.75c+0.50c=1.25c\,,$$
and times that sum by the speed of light,
$$1.25(3*10^8)=3.75*10^8\mathrm{,}$$
to calculate the velocity of the canister relative to an observer from Earth:
$$\vec v_{ab} = 3.75*10^8\,\mathrm{\frac{m}{s}\\}\mathrm{.}$$
Example 2
Now let's try a problem with the subtraction of relative vector velocities.
Everything is the same for this problem, except that the canister is getting ejected away from the Earth rather than towards it. Therefore, we add our velocities as we did before; the only difference is that the velocity of the canister is negative. Recall our equation for relative velocities:
$${{\vec v}_a}_b = ({{\vec v}_a}_d + {{\vec v}_d}_b)\,.$$
Now, the velocity of the canister is negative \(0.75c\). Therefore, its velocity relative to the spaceship is:
$${{\vec v}_a}_d= -0.75c\,.$$
The spaceship is already moving at \(0.50\) times the speed of light, therefore, its velocity relative to an observer from the earth is \(0.50\) times the speed of light:
$${{\vec v}_d}_b = 0.50c\,.$$
We then add our velocities together to figure out how an observer would perceive the canister's velocity from the earth. Substituting our values into our equation gives
$${{\vec v}_a}_b = -0.75c+0.50c$$
and adding those velocities,
$$0.75c+0.50c=-0.25c\mathrm{,}$$
then multiplying them by the speed of light,
$$-0.25(3*10^8)=-7.5*10^7\mathrm{,}$$
yields our answer:
$${{\vec v}_a}_b = -7.5*10^7\,\mathrm{\frac{m}{s}\\}\mathrm{.}$$
Usually, you cannot exceed the speed of light. To get the actual velocity of the canister, you would need to use a relativistic velocities formula:
$$u=\frac{v+u'}{1+\frac{vu'}{c^2}\\}\\\mathrm{.}$$
But that is beyond the scope of this article. Therefore, we ignored the laws of relativity for the above examples and treated them as relative velocities.
Frame of Reference - Key takeaways
- A frame of reference is a system with objects and characteristics that we design to tackle a particular physics problem.
- By choosing a reference frame, one is also choosing to define the direction and magnitude of the parts of the physical system measured by an observer in that frame of reference.
- Measurements within a given reference frame may be converted to measurements within another reference frame.
- The phrase, "an object in motion will remain in motion," identifies an inertial reference frame. Any frame of reference with no external net force acting on it is an inertial reference frame because its acceleration will remain constant relative to all other frames.
- The acceleration of any object is the same as measured from all inertial reference frames.
- Non-inertial reference frames address the second part of Newton's first law: "unless acted upon by an outside force." These frames of reference involve external forces, meaning the reference frame has a non-zero acceleration.
- An observed object's velocity is found by adding the observed object's velocity and the observer's velocity. This is done through the addition or subtraction of vectors: $${{\vec v}_a}_b= ({{\vec v}_a}_d+ {{\vec v}_d}_b)\mathrm{.}$$
References
- Fig. 3 - Space Sun Star (https://pixabay.com/illustrations/space-sun-star-earth-planet-orbit-1251733/) by GuillaumePreat (https://pixabay.com/users/guillaumepreat-1602476/) is licensed by Pixabay License (https://pixabay.com/service/license/)
Explanations 
Exams 
Magazine 
