Physics Tutorial: Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws

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In this Physics tutorial, you will learn:

  • What is an inertial frame of reference?
  • What does the Galilean Principle of Relativity say about motion?
  • What are the Galilean transformations? What are the equations expressing them?
  • What do the Einstein Postulates say on relativity?
  • What is an event is relativity?
  • What do the spacetime coordinates in the Newtonian System represent?
  • How do the Newton's Laws apply in inertial frames of reference?
  • What are non-inertial frames of reference?
  • What are some inertial systems of reference we can choose when studying the motion of particles? Which is the best one?

Introduction

Suppose you are lying on a bed to take a rest. Are you moving in this case? Are you sure? Where do you base the idea that you are moving/not moving?

What can you say about the situation in which you are travelling by car and see the trees around the street moving backward? Do they really move backward? Are they moving in any way?

Is the Earth moving? What about the Sun? Solar system? Galaxies? The universe?

When do you have the sensation that the time elapses faster: during the working days or during the weekend? Why? Does the time really elapses at different paces in this case?

In this tutorial, we will discuss about a phenomenon known as "relativity", which implies the occurrence of an event at different paces, depending on the frame of reference we use to describe the event. Once, this topic was a taboo, until Einstein put it in scientific base.

In scientific terms, relativity is the field of study that measures events (things that happen): where and when they happen, and by how much any two events are separated in space and in time. In addition, relativity deals with transforming such measurements (and measurements of energy and momentum as well) between reference frames that move relative to each other. (From here, it brings the name relativity.)

Inertial Frames of Reference. Galilean Relativity

In physics, a frame of reference is an arbitrary set of axes used to determine the position of an object or the physical laws that govern its motion. When explaining the motion of objects in Section 3, the first thing we did, was appointing an (fixed) origin and a positive direction to any possible motion and then, calculating the corresponding quantities (especially any change in them) accordingly.

Thus, for example, if we say: "An object is at x0 = 2m and then it moves at x = 7m" we mean: "The object was initially 2 m away from the fixed origin in the positive direction and then, it moved for other 5 m still in the positive direction and finally reached the position x = 7m. Therefore, the displacement of the object is Δx = x - x0 = 7m - 2m = 5m."

However, as pointed out above, these values are obtained when we assume the origin as fixed (non-moveable). If the moving object was a car and one was inside the car, he would not agree with our opinion about the above numbers (the change in coordinate). He would probably choose any point inside the car (for example his seat) as origin and measured any change in coordinates accordingly. In this case, this person would get the values x0 = 0 and x = 0 and for the corresponding displacement: Δx = x - x0 = 0m - 0m = 0m.

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In the above example, there are two frames of reference. The first frame is connected to the observer at rest (the person who is standing up outside the car), while the other is connected to the observer in motion (the person sitting inside the car). If they are looking to each other, they obtain the same value for the displacement. Thus, for the observer at rest, the car (and the person inside it), moves by 5 m due right. The same thing occurs to the person who is sitting inside the car; it seems that the observer at rest is moving by 5 m due his right as well, when he looks from the car window.

Thus, in the above example, we have two distinct frames of reference (i.e. places that are moving at constant speed relative to each other): one is at rest (as it looks to the observer at rest) and the other is in synchronized motion to the moving observer. It is obvious that Newton's Laws of Motion can be applied in both the above reference frames with the proper corrections. Thus, if the car is moving at constant speed v, there is no acceleration detected by any of observers, as for the first observer (the one at rest), we have:

a1 = v - v0/t = v - v/t = 0 - 0/t = 0/t = 0

and for the one who is sitting inside the car (v = 0), we have:

a2 = v - v0/t = 0 - 0/t = 0/t = 0

The above example involves a one-dimensional motion but we can generalize it in three dimensions as well. In this way, we obtain the definition (and explanation) of "inertial frames of reference":

Inertial frames of reference are three-dimensional coordinate systems, which travel at constant velocity. In such frames, an object is observed to have no acceleration when no forces are acting on it. If a reference frame moves with constant velocity relative to another inertial reference frame, it represents an inertial reference frame as well. There is no absolute inertial reference frame; this means there is no state of velocity that is special in the universe. All inertial reference frames are equivalent. One can only detect the relative motion of one inertial reference frame to another.

The term "inertial" derives from the concept of "inertia" discussed in the Newton's First Law of Motion, which implies that an object moves at constant velocity unless an unbalanced force acts on it.

The idea of relativity of motion was not invented by Einstein (who is the first person that comes in our mind when speaking about relativity) but had instead existed in scientific circles long before him. It deals with Newtonian mechanics as well. The key point of relativity concept is that the results of experiments done in any inertial frame will be the same; i.e., one will not be able to determine by some experimental means what frame is at "absolute rest" and which are moving, as all motion is relative. As explained above, an inertial frame is one in which Newton's laws of motion are satisfied. These laws prescribe accelerations, not velocities, so a frame that is moving at constant relative velocity with respect to an inertial frame is an inertial frame as well.

The non-priority of an inertial frame of reference to another inertial one constitutes the Galilean Principle of Relativity, formulated in 1635, long before Einstein generalized the concept of relativity by including non-inertial reference frames as well.

Mathematically, the Galilean Principle of Relativity expresses the invariance of mechanical equations with respect to transformations occurring in the coordinates of moving points (and time) when there is a transition from one inertial system into another. This means we have four variables included in such situations: the three spatial coordinates x, y, z and the time t.

Galilean Transformations

Let's consider an inertial reference frame S which we assume at rest and another inertial reference frame S' moving at constant velocity v in respect to S. An object is moving horizontally at velocity v in the system S in the moving direction of S'. It is obvious that this object will be at rest in respect to the inertial system S' as they are moving at the same velocity and direction (v' = 0).

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We want to know how to determine the coordinates in S' when we know them in frame S. In the figure above, in the S frame the object is moving and the {x, y, z} axes are fixed. When we transform to the S' coordinate system (so that {x', y', z'} are at rest), it now looks like the object has velocity v' = 0 and that the old axes {x, y, z} are moving with velocity - v in the NEGATIVE x' direction. In this case, "v" appears as the relative velocity of the PRIMED coordinate system {x', y', z'} or S', compared to the UNPRIMED coordinate system {x, y, z} or S.

To determine the object's coordinates in the inertial frame, S', when we know its coordinates in the original inertial frame, S, we employ the Galilean space and time transformations. If S' has a velocity relative to S so that v' = 0, then we have:

x' = x + v ∙ ty' = y
z' = z
t' = t

In Galilean transformations, time is the same in all inertial frames.

If the objects moves in three dimensions, its velocity v contains three non-zero components vx, vy and vz. Therefore, the Galilean transformations if the inertial frame of reference S' moves at the same velocity v, become:

x' = x + vx ∙ t
y' = y + vy ∙ t
z' = z + vz ∙ t
t' = t

In the example discussed in the previous paragraph, the observer at rest is thought as being at rest in respect to the inertial system S, while the passenger inside the moving car is at rest in respect to the system S' which has its origin at any point inside the car and which is moving at constant velocity v in respect to the system S. This moving velocity of the inertial system S' represents the car's velocity. Let's see an example to clarify this point.

Example 1

Two cars A and B are moving toward each other at constant speeds vA = 20 m/s and vB = 10 m/s respectively as shown in the figure.

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The car A is initially 400 m on the left of the tree, while the car B is initially 500 m on the right of the tree. Find the initial position of the tree is you choose:

  1. Position of the tree as origin of the inertial system of reference
  2. Position of the car A as origin of the inertial system of reference
  3. Position of the car B as origin of the inertial system of reference

In addition, find:

  1. Equations of motion for each car in each inertial reference system
  2. Position of the meeting point for the two cars in each inertial reference system

Solution 1

  1. If we take the position of the tree as origin of the inertial reference system and assume as positive the left-to-right direction in the figure, we obtain for the initial position x0T of the tree (and also the position of tree at any instant):
    x0T = xT = 0 m
  2. If we take the position of car A as origin of the inertial reference system and assume the positive direction due right of the figure, we obtain for the initial position of tree:
    x0T = + 400 m
  3. If we take the position of car B as origin of the inertial reference system and assume the positive direction due right of the figure, we obtain for the initial position of tree:
    x0T = -500 m
  4. If we choose the tree position as origin of reference frame, we have for the initial position of car A: x0A = - 400 m and for the speed of car A: vA = + 20 m/s. Therefore, since the general equation of motion for the car A at any reference frame is
    xA (t) = x0A + vA ∙ t
    then, the equation of car A when the tree is chosen as origin, is
    xA (t) = -400 + 20 ∙ t
    Using the same approach, we find for the car B when the tree is chosen as origin: x0B = + 500 m and vB = -10 m/s. Therefore, since the general equation of car B for this reference frame is
    xB (t) = x0B + vB ∙ t
    we obtain after substitutions
    xB (t) = 500-10 ∙ t
    When the car A is chosen as origin of the inertial reference system, then the car A is at rest, and moreover, the initial position x'0A is also zero. Thus, since
    x'A (t) = x'0A + v'A ∙ t
    we obtain after substitutions
    x'A (t) = 0 + 0 ∙ t
    This means the car A will always be at position 0 for this inertial reference frame.
    As for the car B when the position of car A is chosen as origin, we have x'0B = + 400 m + 500 m = + 900 m and the velocity of the car B in respect to the car A is
    v'B = vB-v'A = -10 m/s-20 m/s = -30 m/s
    Therefore, since the general equation of car B when the car A is chosen as origin is
    x'B (t) = x'0B + v'B ∙ t
    we obtain after substituting the known values:
    x'B (t) = 900-30 ∙ t
    On the other hand, when the car B is chosen as origin of the inertial reference frame, we have for the initial position of car A:
    x''0A = -400m-500m = -900 m
    and for the velocity of car A:
    v''A = vB-v'A = 0-(-30 m/s) = 30 m/s
    Hence, since the general equation of the car A when the car B is chosen as origin is
    x''A (t) = x''0A + v''A ∙ t
    we obtain after substituting the known values:
    x''A (t) = -900 + 30 ∙ t
    As for the car B, it is considered always at rest if the car B itself acts as origin. Thus, we have
    x''0B and v''B = 0
    Thus, we obtain for the equation of car B in this reference frame:
    x''B (t) = x''0B + v''B ∙ t
    or
    x''B (t) = 0 + 0 ∙ t
    Again here, the car B is considered at rest during the entire process if the origin of the inertial reference frame is in this car.
  5. The two cars meet if their coordinates are equal. The time is also taken the same as the two events are simultaneous. Thus, if we choose the tree as origin, we must write for the meeting point:
    xA (t) = xB (t)
    Thus, substituting the expressions obtained in (d), we obtain
    -400 + 20 ∙ t = 500 - 10 ∙ t
    20 ∙ t + 10 ∙ t = 500 + 400
    30 ∙ t = 900
    t = 30 s
    Thus, using any of equations of motion for this inertial system of reference (for example that of the car A), we obtain for the meeting position in respect to the tree:
    xA (t) = -400 + 20 ∙ t
    = -400 + 20 ∙ 30
    = -400 m + 600 m
    = + 200m
    This means the two cars meet 200 m on the right of the tree.
    The same result could also have been obtained if we used the equation of the car B. Thus,
    xB (t) = 500 - 10 ∙ t
    = 500 - 10 ∙ 30
    = 500 m - 300 m
    = 200 m
    Moreover, we obtain the same result for any car taken as origin at any inertial reference frame. Let's check this claim. Thus, since the time is equal for each inertial system of reference, we have in case when we take the car A as origin:
    x'A (t) = 0 + 0 ∙ t
    = 0 + 0 ∙ 30
    = 0
    and
    x'B (t) = 900-30 ∙ t
    = 900 - 30 ∙ 30
    = 900 m - 900 m
    = 0
    This result is understandable; the position of meeting point must correspond to the origin of reference frame when the car A is chosen as inertial reference system because the meeting point always correspond to the position of car A. The same this can be said when we choose the car B as origin. In this case, we have
    x''A (t) = -900 + 30 ∙ t
    = -900 + 30 ∙ 30
    = -900 m + 900m
    = 0
    and
    x''B (t) = 0 + 0 ∙ t
    = 0 + 0 ∙ 30
    = 0

Remark! When any of cars is chosen as origin and we are interested to find their meeting point, the tree is not involved in calculations as it is out of interest for the event. This is why the tree position is not mentioned in the last two inertial reference systems.

The Postulates

Regardless of the work done by Galileo and other scientists in the field of relativity, it is Einstein the person who is credited for putting the concept of relativity on scientific basis. Initially, Einstein formulated the Special Theory of Relativity, which is valid only for inertial frames of reference (in which the Newton's Laws of Motion can be applied, as explained earlier) and later, he generalized his findings by formulating the General Theory of Relativity, which is also valid for non-inertial frames that undergo gravitational acceleration (this is a complex theory which goes beyond the scope of this section).

Sometimes, findings and discoveries in science are difficult to be made in a straightforward way, through direct measurements and experiments. Therefore, indirect methods based on assumptions are often used in such cases to prove a certain theory. There are some basic concepts that are taken as true without proof and then, the new theory is built based on them. Such concepts are known as "postulates". In other words, a postulate is a thing suggested or assumed as true as the basis for reasoning, discussion, or belief.

Einstein based his Special Theory of Relativity upon two postulated:

  1. The laws of physics are the same and can be expressed in their simplest form in all inertial frames of reference (we discussed this point earlier in this tutorial). This is known as the relativity postulate. The laws of physics mentioned in this postulate include only those that satisfy this postulate.
  2. The speed of light in vacuum has the same value c in all directions and in all inertial reference frames. This is known as the speed of light postulate.

In other words, the second postulate implies that in universe, there is an ultimate speed c, which is the same in all directions and in all inertial frames of reference. Light is the only known thing that travels at this ultimate speed. No entity that carries energy or information can exceed this limit. Moreover, no particle that has mass can actually reach the ultimate speed c, no matter how much or for how long that particle is accelerated.

Precise measurements using modern devices have given the value c = 299,792,458 km/s for the speed of light in vacuum. We often round this value to 300,000,000 km/s or 3 × 10-8 m/s. However, in this chapter (section) we will use the exact value to describe the speed of light in vacuum.

Both postulates have been tested many times but they have resulted always true. The existence of a limit to the speed of accelerated electrons was shown in an experiment carried out in 1964 by W. Bertozzi, who accelerated electrons to various measured speeds and through an independent method, he measured their kinetic energies. He found that as the force on a very fast electron is increased, the electron's measured kinetic energy increases toward very large values but its speed does not increase appreciably. The graph showing the relationship between the speeds of electrons and their kinetic energy is shown below.

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Measuring an Event. The Spacetime Coordinates in the Newtonian System and the Newton's First Law

An event is an occurrence, i.e. something that happens in a certain region of space in a given time. An event is fully described when we know four parameters of it: the three space coordinates (x, y, z) and the time (t). These are known as the spacetime coordinates because in relativity, space and time are entangled with each other.

As explained earlier in this tutorial, a given event may be recorded by any number of observers, each of them being in a different inertial reference frame. In general, different observers will assign different spacetime coordinates to the same event, although an event does not "belong" to any particular inertial reference frame. An event is just something that happens, and anyone in any reference frame may detect it and assign spacetime coordinates to it.

Newton believed in the existence of an absolute space and an absolute time entirely independent from physical processes and from the existence of material objects. In simple words, this belief consisted on the idea that "even if we removed all objects from the universe, the space and time would still remain in it". According to Newton, there exists an eternal and homogenous space (no time instant is privileged over the others). All events occur in this "super vessel" with infinite dimensions and in this time with infinite duration. These events range from the simplest (mechanical processes) to the most complicated, including the life and evolution processes.

The absolute and infinite space of Newton is a three-dimensional space, while his absolute time is one-dimensional. To illustrate this idea, let's choose a whatever point in the absolute space which we call "origin, O" and three perpendicular axes which we call the "Cartesian axes" (x, y, z). Thus, an absolute coordinate system S0 (O, x, y, z) is obtained. Now, whatever point distinct from S0 has the coordinates (x, y, z) as shown in the figure below.

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Let's assume a material point (particle) totally isolated in the above three-dimensional space (after assuming to have removed all the other objects in this space). In such condition, the particle has two options: either to stay in the same point of the absolute 3-D space to move in a linear path by changing continuously its coordinates according the rate of the absolute time flow. This is nothing else but the famous "Law of Inertia", postulated initially by Newton. This type of motion is known as "standard motion". The particle in question does not interact with anything else, given that it is isolated in the absolute space. For this reason, it is called a "free particle". We say "a free particle moves in standard mode in the absolute system of space coordinates" and this system is called the "absolute inertial system of reference." As soon as we create the conditions to have a system of reference S which is inertial to the absolute system of reference S0, then every other system that moves in standard mode to this absolute system is inertial as well. Look at the figure.

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In the above figure, the point P can be represented though two sets of coordinates: (x0, y0, z0) in the absolute system of reference S0 and (x, y, z) in the inertial system of reference S. Despite the fact that the point P is the same, it has two pairs of coordinates because two inertial systems of reference are being considered, where the inertial system S moves at constant velocity v in respect to the absolute system S0. Therefore, the conditions for the Newton's First Law of Motion are met because this law includes either the objects at rest or those moving at constant velocity (if no force is acting on the object, it either remains at rest or moves at constant velocity).

In the above example, we can consider the situation in two viewpoints. If the absolute system S0 is taken as a system of reference, the point P remains at rest. On the other hand, if we take the system S as a system of reference, the point P is considered in uniform motion, despite it is not moving, because its coordinates change due to the shift of the reference system.

Newton's Second and Third Law and Systems of Reference

If a particle in an inertial system of reference is accelerated, i.e. if it does not perform a standard motion, we say the particle interacts,/b>. A force F - which measures the intensity of interaction - acts on the particle. On the other hand, the acceleration a the particle experiences, gives the intensity of particle's deflection from the standard motion. As explained in Section 4, the relationship between these two quantities is given by the Newton's Second Law of Motion.

a = F/m

Hence, it is clear that the acceleration a particle experiences due to the action of a force is proportional to the force itself. This statement represents the Newton's Second Law of Motion expressed in words. Here, 1/m is the coefficient of proportionality, which turns the given proportion into equation and the mass itself expresses the inertia, i.e. the tendency of an object to resist to any change in its state of motion.

Newton's Third Law of Motion (the action-reaction principle) completes the framework of Newtonian Dynamics. It says: "For every action, there is an equal-size but opposite reaction".

In this way, it is clear that when the net force acting on an object is zero, the object moves in standard mode (i.e. at constant velocity). Since it is practically impossible to find any free particle in Earth conditions, we can obtain inertial systems of reference only if the net force on an object is zero, as this situation is more realistic.

If we neglect some factors such as air drag, friction, etc., the system connected to the Earth can be considered more or less as a good inertial system of reference. A flying airplane when no air turbulences are present, is also a good inertial system of reference as well as the system connected to the Sun. However, the best model of an inertial system is the one connected to a spacecraft moving into the interstellar space with the engines tuned off.

Summary

In physics, a frame of reference is an arbitrary set of axes used to determine the position of an object or the physical laws that govern its motion.

Inertial frames of reference are three-dimensional coordinate systems, which travel at constant velocity. In such frames, an object is observed to have no acceleration when no forces are acting on it. If a reference frame moves with constant velocity relative to another inertial reference frame, it represents an inertial reference frame as well. There is no absolute inertial reference frame; this means there is no state of velocity that is special in the universe. All inertial reference frames are equivalent. One can only detect the relative motion of one inertial reference frame to another.

The term "inertial" derives from the concept of "inertia,/b>" discussed in the Newton's First Law of Motion, which implies that an object moves at constant velocity unless an unbalanced force acts on it.

The non-priority of an inertial frame of reference to another inertial one constitutes the Galilean Principle of Relativity, formulated in 1635, long before Einstein generalized the concept of relativity by including non-inertial reference frames as well.

Mathematically, the Galilean Principle of Relativity expresses the invariance of mechanical equations with respect to transformations occurring in the coordinates of moving points (and time) when there is a transition from one inertial system into another. This means we have four variables included in such situations: the three spatial coordinates x, y, z and the time t.

To determine the object's coordinates in the inertial frame, S', when we know its coordinates in the original inertial frame S, we employ the Galilean space and time transformations. If S' has a velocity relative to S so that v' = 0, then we have:

x' = x + v ∙ t
y' = y
z' = z
t' = t

In Galilean transformations, time is the same in all inertial frames.

Einstein based his Special Theory of Relativity upon two postulated:

  1. The laws of physics are the same and can be expressed in their simplest form in all inertial frames of reference (we discussed this point earlier in this tutorial). This is known as the relativity postulate. The laws of physics mentioned in this postulate include only those that satisfy this postulate.
  2. The speed of light in vacuum has the same value c in all directions and in all inertial reference frames. This is known as the speed of light postulate.

The second postulate implies that in universe, there is an ultimate speed c, which is the same in all directions and in all inertial frames of reference. Light is the only known thing that travels at this ultimate speed. No entity that carries energy or information can exceed this limit. Moreover, no particle that has mass can actually reach the ultimate speed c, no matter how much or for how long that particle is accelerated.

Precise measurements using modern devices have given the value c = 299,792,458 km/s for the speed of light in vacuum.

An event is fully described when we know four parameters of it: the three space coordinates (x, y, z) and the time (t). These are known as the spacetime coordinates because in relativity, space and time are entangled with each other.

Newton believed in the existence of an absolute space and an absolute time entirely independent from physical processes and from the existence of material objects. If a particle moves uniformly in deep space, this is known as "standard motion". The particle in question does not interact with anything else, given that it is isolated in the absolute space. For this reason, it is called a "free particle" and the system in which this particle is moving is called the "absolute inertial system of reference." Every other system that moves in standard mode to the absolute system of reference is also inertial.

If an inertial system S moves at constant velocity v in respect to the absolute system S0, then the conditions for the Newton's First Law of Motion are met. This is because this law includes either situations where the objects at rest when they move at constant velocity (if no force is acting on the object, it either remains at rest or moves at constant velocity).

If a particle in an inertial system of reference is accelerated, i.e. if it does not perform a standard motion, we say the particle interacts. A force F - which measures the intensity of interaction - acts on the particle. On the other hand, the acceleration a the particle experiences, gives the intensity of particle's deflection from the standard motion. The relationship between these two quantities obeys the Newton's Second Law of Motion

a = F/m

Thus, the acceleration a particle experiences due to the action of a force is proportional to the force itself. This statement represents the Newton's Second Law of Motion expressed in words. Here, 1/m is the coefficient of proportionality, which turns the given proportion into equation and the mass itself expresses the inertia, i.e. the tendency of an object to resist to any change in its state of motion.

The Newton's Third Law of Motion (the action-reaction principle) completes the framework of Newtonian Dynamics. It says: "For every action, there is an equal-size but opposite reaction".

In this way, it is clear that when the net force acting on an object is zero, the object moves in standard mode (i.e. at constant velocity). Since it is practically impossible to find any free particle in Earth conditions, we can obtain inertial systems of reference only if the net force on an object is zero, as this situation is more realistic.

An inertial connected to a spacecraft moving into the interstellar space with the engines tuned off represents the best model of an inertial system.

Physics Revision Questions for Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws

1. A truck is moving at 126 km/h due East and a car is travelling at 30 m/s due West. Both objects are moving at constant velocity. The initial distance between the vehicles is 1.2 km. Which is the equation of motion by which the car approaches the truck if a traffic light - which is 300 behind the truck - is chosen as origin of the inertial frame of reference? Take the West-to-East direction as positive.

  1. x(t) = 1200 - 65t [in metres]
  2. x(t) = 1500 - 65t [in metres]
  3. x(t) = 1500 - 156t [in metres]
  4. x(t) = 900 - 156t [in metres]

Correct Answer: B

2. A passenger is walking at 2m/s towards the end of a bus which is moving at 72 km/h. What is the relative velocity of the passenger to the ground if the moving direction of the bus is taken as positive?

  1. 74 km/h
  2. -70 km/h
  3. 22 m/s
  4. -18 m/s

Correct Answer: D

3. Which of the following is NOT an inertial frame of reference?

  1. The Earth
  2. The Sun
  3. A car moving at constant velocity
  4. An object falling freely

Correct Answer: D

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