Kinematics: _{Speed in one dimension}

Introduction

Human beings love a race. More than 5 billion people followed the athletic events at the Paris Summer Olympics in 2024 where international athletes competed on foot, in the water, in boats, on bicycles, and on climbing walls to see who was the fastest. In Formula One racing, ten teams compete to develop the fastest car, like the one shown in Figure 1. More than 750 million people follow the series, watching as drivers push their cars to speeds that reach 230 miles per hour (mph).

When we talk about a car traveling at 230 mph, we are describing its speed. A runner or a car wins a race because it travels at the fastest speed. Yet speed cannot be measured directly. Most approaches to determining speed measure the time required to cover a known distance—this is how the speed of a runner or car is determined in a race. Instruments like speedometers or radar guns actually measure a force or a frequency change that depends on the speed of an object. These are all useful techniques in our daily lives and help determine how fast a car is going or who won the 100-yard dash.

But how do we measure things that are extremely fast or extremely slow? How do physicists know that light travels at 299,792,458 meters per second (m/s)? How do climate scientists know that sea levels are rising at 2.3 millimeters per year? To answer these questions, it’s helpful to define speed and explore early attempts to measure and describe it.

Defining speed

In physics, speed is defined as the distance traveled over a period of time. It is a rate of motion expressed mathematically with the equation:

This is a linear equation. It tells us that to calculate speed, we divide the distance an object travels (distance traveled) by the amount of time it takes to cover that distance (time elapsed). You can read more about rates and how to calculate them in our Linear Equations module.

[This equation is often written in shorthand notation using variables. We will use the variable x to represent distance and t to represent time. The symbol \( \Delta \) is used to indicate the change to a variable, so we will use \( \Delta x \) to represent distance travelled and \( \Delta t \) to represent time elapsed. The variable \( \upsilon \) is used to represent speed. So, the equation may be written as:]

Speed is measured and expressed with different units depending on the context:

• A car speedometer measures speed in kilometers per hour (kph) or miles per hour (mph).
• Physicists typically use meters per second (m/s) in their calculations.
• Geologists use millimeters per year (mm/yr) to describe rates of plate motion.

Regardless of the context, all of these units are a ratio of a unit of distance (miles, meters, etc.) per a unit of time (hour, year, second). Whenever you see a unit with the appearance of a distance per time, you know that you are seeing a rate of motion, or a speed. Using that definition and mathematical expression, you can easily calculate your speed during a drive along a highway. There are markers along the highway showing the distance, like marks on a ruler (Figure 2).

Suppose that you drive past mile marker 106 at noon (12:00 pm) and you pass mile marker 166 at 1:00 pm. To calculate your speed, you would use the following steps:

When calculating this value, you assume the speed is constant to simplify a complex situation. In reality, your car’s speed may fluctuate for many reasons, like traffic or curves in the road, and the car’s speedometer would reflect that change. When you calculate the speed based on the total distance and total time, you are finding the car's average speed over that period. In this case, you treat the motion as having a constant rate to simplify the calculations. Constant-speed motion provides a useful model for understanding how objects move in many real-world scenarios, even if the actual motion over time is more variable.

You may have heard the word “velocity” used interchangeably with speed. However, these two terms have distinct meanings in physics. Speed is a scalar quantity, which means it only has magnitude (how fast something is moving). Velocity, on the other hand, is a vector quantity, which includes both the magnitude (speed) and the direction of motion. Along a straight highway, both the speed and the velocity will be constant. But when the road is curved, the velocity will change even when the speed stays the same. In this module, we are focused on the scalar quantity of speed.

Early attempts to measure the speed of light

Measuring the speed of something requires that it be in motion. In the early 1600s, most people thought it was impossible to measure the speed of light because they did not see light move. Instead, they perceived light as instantaneously everywhere. However, Italian astronomer and physicist Galileo Galilei was unconvinced that this was the case. Galileo, often referred to as the father of classical physics, was born in Pisa, Italy in 1564. He is primarily known as an astronomer, who used the high quality telescopes he produced to make observations of the solar system. His observations of the moons of Jupiter (now known as the Galilean moons) and the phases of Venus provided strong evidence in favor of the Sun-centered (Copernican) model of the solar system. In 1638, his final scientific writing, “Dialogues concerning two new sciences”, addressed the nature of motion using the tools of mathematics, creating the field of kinematics.

In this work, Galileo designed an experiment to measure the speed of light. His design was straightforward, using the same process we used earlier to calculate speed of the car on the highway: dividing a known distance by the time it took light to cover that distance.

Here’s how Galileo’s experiment worked:

One observer (Observer 1) stands on a hill about a mile away from a second observer (Observer 2), who is standing on another hill (Figure 3). They can see each other, though each appears very small to the other. Each holds a lit lantern that they cover. Then:

• Observer 1 briefly uncovers his lantern, sending a beam of light toward Observer 2, and he starts a timer at the same time;
• As soon as Observer 2 sees the light, he uncovers his lantern, sending a beam back to Observer 1;
• When Observer 1 sees the light from Observer 2’s lantern, he stops the timer, recording the time it took the light to travel the round trip, or twice the known distance between the two observers.

By knowing the distance between the two hills and the measured time, Galileo hoped to calculate the speed of light and show that it was not instantaneous.

Even though he was using the best timing methods available at the time, Galileo was unable to measure such a short period of time. Light travels at such a high speed that, when measured over the distance of a mile, it appears instantaneous by human standards. Human reaction times are much slower. As a result, Galileo found the experiment inconclusive: Either the speed of light was too fast to be measured with this method or it travelled from place to place in no time at all. As is often the case in science, a definitive answer for the speed of light would have to wait for a different kind of experiment.

Representing the relationship between position and time

Galileo used drawings and words to represent his findings about motion and change in position over time. Today, motion with a constant speed is more often represented using algebraic equations, tables of values, or graphs.

The algebraic equation that relates speed (represented by the variable v) to position and time is:

This equation states that \( x \) (the position of the object at time \( t \)) is calculated by adding the initial position (x₀) to the distance traveled, which is found by multiplying the speed (\( v \)) by the time elapsed (\( \Delta t \)). This is the same relationship as our definition for speed, just written in a different form as can be seen by manipulating the equation:

There are many ways of representing motion. We will consider the different ways of representing a walk that Galileo takes across his room.

Galileo starts a distance of 2.0 m away from the left-hand wall of the room and walks quickly to the right until, three seconds later, he reaches a distance of 6.5 m away from the left-hand wall of the room.

This verbal description helps us to envision what is happening. We can picture Galileo taking quick, large steps from left to right across a wide room.

Table 1 represents the same motion, but focuses our attention on Galileo’s measured position at several instants in time. Here, we can read where Galileo was at specific moments in between the start and end of the walk as described.

In the graph shown in Figure 4, time is plotted on the horizontal axis and position is plotted on the vertical axis. The same data are shown as in the table, but we can use a line that fits the data to infer where Galileo was at all moments in time, including moments that are not shown in the table.

We can solve this equation for v, finding that it is equal to 1.5 m/s.

In fact, we can use any of these three representations to find Galileo’s speed. In the verbal representation, the speed can be determined from the starting and ending positions and the time.

As before, we can use the distance travelled (6.5 m - 2.0 m = 4.5 m) and the time (3 seconds) to calculate an average speed of 1.5 m/s (4.5 m/3 s = 1.5 m/s).

The data in Table 1 indicate that Galileo moves at a constant speed because his change in position during each one-second time interval is the same: 1.5 meters. We can therefore determine the average speed during each second as 1.5 m/s because the position increases by 1.5 meters each second.

In the graph shown in Figure 5, we see the same constant-speed motion because the measured points all fall along a line with a constant slope. On a position vs. time graph, the slope of the line is the speed. Figure 5 is the same as Figure 4, but we have added a green line to show the time elapsed between 1 and 3 seconds. Over this time, Galileo moves a distance of 3 meters, shown by the blue line. Calculating the slope as the ratio of these is the same calculation as finding the speed.

The graph representation also helps us understand the algebraic function that shows the same motion. The equation of a line has two parameters: the slope (m) and the y-intercept (b). The form is the same as the equation for constant-speed motion. We can match the parameter m with the v in the equation for constant-speed motion. Similarly, we can match the parameter b with the x₀ in the equation for constant-speed motion.

Finding a speed from collected data

Not all motion is as simple to imagine as Galileo walking across a room. Often, position data is difficult to collect, especially when the speed is very fast or slow. Additionally, when measuring an object’s position at different times, the data points may not fall perfectly along a line. So, how do scientists approach finding a speed for more complex motion? As an example, let’s look at the phenomenon of sea level rise.

When scientists want to study a phenomenon as complex as the motion of the sea’s surface, there are many considerations. Over what timescale are we interested in the motion? Waves change the surface level from over seconds and tides change the surface level over hours. Even the motion of tectonic plates can change the sea’s surface level over thousands of years! How can we determine the “average” height over all the planet’s oceans when we can only measure at one point at a time?

The graph in Figure 6 shows global mean sea level since 1900. Each data point on the graph represents hundreds of measurements from around the world, corrected for all the factors listed above and averaged together.

You might first notice the jagged up and down of the dark blue line, showing increases and decreases of around 10 mm over a few years. You can also see a general trend of the sea level rising over time. In 2020, global mean sea level was 208 mm higher than it was in 1900.

The data in Figure 6 do not produce a perfectly straight line because sea level is not changing at a constant rate. However, we can use the beginning and end points to calculate an average speed of the sea’s surface.

This unit of speed (millimeters per year, mm/year) is clearly very slow, making a direct comparison to Galileo’s walking of meters per second (m/s) difficult. To make that comparison, we can use dimensional analysis to convert the unit of meters per second to millimeters per year (see our Unit Conversion module).

This analysis tells us that the speed of sea level rise is about a trillion times slower than the speed at which people walk or drive cars; it is the equivalent of increasing the depth of the ocean by one atom each second. In addition to all the complexities mentioned above, measuring such a slow speed is only possible by making measurements over very long periods of time.

Using appropriate distances to measure the speed of light

Similar to how the speed of sea level rise must be measured over long periods of time, the speed of light must be measured over very long distances. Astronomers Giovanni Cassini and Ole Rømer first successfully measured a speed of light in the late 1600s, using a dramatically larger distance than in Galileo’s experiment. Instead of a flash of lamp-light on a hill a mile away, Cassini and Rømer studied the eclipse of Jupiter’s moon Io, 760 million kilometers away. Io is one of the Galilean moons, named because of their first observation by Galileo who suggested using these moons as a clock observable from anywhere on Earth. This is precisely what Cassini hoped to achieve with his careful measurements of the eclipse.

As astronomers, Cassini and Rømer knew that the distance between Earth and Jupiter was not constant. Figure 7 shows how this distance changes throughout the year as the two planets orbit the Sun. Jupiter moves much more slowly than Earth, orbiting the Sun once every 12 Earth years. The left panel shows the shortest distance between the planets. In the middle and right panels, the distance between the two increases.

Io orbits Jupiter every few Earth days, and is eclipsed (for an observer on Earth) when it passes behind Jupiter on each orbit. Cassini’s observations showed that the period of Io’s orbit was close to, but not consistently, 2550 minutes. When Earth was closest to Jupiter, the eclipse would be observed 11 minutes ahead of the predicted time. When Earth was farthest from Jupiter, the eclipse was observed 11 minutes after the predicted time. Cassini wrote in 1676, “This inequality appears to be due to light taking some time to reach us from [Io]; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of [Earth’s] orbit.” The middle panel of Figure 7 shows the “half-diameter” (or radius) of Earth’s orbit and the 11-minute delay.

Rømer thought that the 11-minute variability of the observed eclipse was a result of Io’s light taking some time to reach Earth. In other words, he attributed the difference in time to a finite speed of light and a difference in distance. Despite his insight and the development of this method for measuring the speed of light, Rømer himself never presented a definitive number to other astronomers.

In 1690, Dutch mathematician and physicist Christiaan Huygens used the radius (the ‘half-diameter’) of Earth’s orbit as 150 million kilometers and the time it takes for light to travel that distance as 11 minutes to calculate the speed of light:

Huygens observed, this is “more than 600,000 times larger than that of sound, which is not at all the same thing as being instantaneous.”

The speed of light today

The value Huygens calculated for the speed of light in the vacuum of space was about 23% lower than our contemporary value. In the following centuries, values for the speed of light would become increasingly accurate and precise with the improvement of clocks and mechanical apparatus.

In 1848, French physicist Hippolyte Fizeau constructed a mechanical equivalent of the Galileo “lamp-light” experiment (see Figure 8). In Fizeau’s apparatus, the gap between the teeth in a gear served as a shutter, letting pulses of light about 1/10000 of a second long shine on a mirror almost 8.6 km away. Finally, there was an experiment where extremely short times could be measured precisely. When they returned, these pulses were blocked by the teeth of the gear, which had rotated to overlap the location of the gaps. Fizeau’s work would overestimate the speed of light by about 5%.

Refinements and extensions of Fizeau’s apparatus improved the accuracy of measurements of the speed of light and taught scientists more about the nature of light. In 1850, Fizeau and physicist Leon Foucault independently measured that light traveled at a different speed in water than in air. In 1851, Fizeau observed that light moves faster through water flowing in the same direction, though not by the amount expected if it behaved in the same way as a boat traveling with a river’s current. Albert Einstein referenced this curious result as being influential in his developing the theory of special relativity, which states that light is observed to travel at the same speed by any observer. In 1887, Albert Michaelson and Edward Morely were looking for the presence of a medium for light to travel through “empty” space. They found the opposite result, providing the first evidence that light does not require a medium to travel.

In time, the values for the speed of light would become so precisely measured that they exceeded the precision to which distance was defined. In 1983, the speed of light through empty space was fixed at 299,792,458 m/s, and the meter was redefined as the distance light travels in 1/299,792,458 of a second.

Today, light and accurate clocks are used to measure distances precisely. The relationship between speed, distance, and time can be expressed in different ways depending on what we want to calculate. By rearranging the equation for speed, we can solve for the distance traveled when an object moves at a constant speed for a certain amount of time:

This relationship is the same as our mathematical definition of speed but allows for different quantities to be calculated. For example, scientists use the Jason-3 satellite (Figure 9) for ongoing measurements of sea level to determine global averages. This satellite operates on the same basic principle as Galileo’s lamp-light experiment. A pulse of radio waves (long-wavelength light) is sent from the satellite and bounces off the ocean’s surface. The time between sending the pulse and its return to the satellite is measured. Knowing the speed of light and the time elapsed means that the distance between the satellite and the surface can be calculated with \(\text{distance traveled} = \text{speed} \times \text{time elapsed} \) .

The speed of light is also a key part of the Global Positioning System (GPS), a collection of near-Earth satellites. These satellites continuously broadcast their location and time in a way that can be measured by GPS receivers on Earth. As with the Jason-3 altimeter, this information is carried by pulses of radio waves that travel at the speed of light. By measuring the delay of the signal from satellites at different locations, a GPS receiver can determine how far away each satellite is. This is similar to Cassini and Rømer’s observations; when the distance to the satellite is larger, the delay for the signal is longer as well. The GPS receiver then uses the distances from the satellites to triangulate your position on the surface of Earth.

Knowing the value for the speed of light, we can understand why Galileo’s technique from 1638 was destined to fail. By rearranging the equation for speed, we can solve for time elapsed when an object travels a distance:

Using a distance of one mile (or 1600 m) between the hills, the time it takes for light to travel back and forth would be:

It’s no wonder Galileo would conclude that light moves instantaneously: This time is about 1/10000 of the typical human reaction time of 0.2s.