Velocity vs. Speed

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Speed and velocity are two closely related quantities, both of which measure an object’s movement relative to time. Speed is fully described by a simple numerical value; velocity is speed with added information about the direction of movement. Using equations developed by English physicist Isaac Newton, simple calculations of speed and velocity are useful in day-to-day activities like planning car trips or tracking the distance covered while jogging.

PRINCIPAL TERMS

• average: in physics, the overall value for a given quantity, obtained by comparing initial and final values of a measurement against another unit or quantity; for instance, average speed compares the total distance traveled to the total time taken to move that distance.
• displacement: the absolute distance between where a moving object begins and where it ends, regardless of the object’s path. Displacement may be equal to or less than distance traveled.
• instantaneous: denotes a measurement taken at a specific point in time.
• rate: the ratio of a unit, such as distance or weight, relative to a period of time.
• scalar: a quantity that is fully described by a numerical value alone, such as speed, length, or mass.
• slope: a line on a graph that indicates the rate of change over time; for instance, on a plot of the distance traveled by an object over time, the slope equals the speed of travel and its shape can convey information about acceleration.
• vector: quantities that require a direction, along with a numerical value, to be fully described, such as velocity or acceleration.

Going Where and How Fast?

Knowing how quickly an object is moving, and in what direction, is very useful. It is so useful that many people have an intuitive sense of these things. Drivers constantly estimate speed and velocity, watching the approach of a nearby car at an intersection to determine whether it is safe to cross, for instance. The formalized versions of these calculations underpin much of transportation and navigation. A car’s onboard GPS uses information about a car’s velocity, the position of its destination, and the total distance of various routes to give arrival estimates; a hiker with a detailed map and accurate compass may do much the same.

Speed and velocity fall under a subfield of classical mechanics known as "kinematics." Kinematics focuses on the motion of objects and can trace its roots back to classical Greek philosophers and mathematicians like Aristotle. The kinematics most used today was defined by English physicist Isaac Newton (1642–1727) in eighteenth century. In the twenty-first century, the most accurate systems of kinematics rely on relativistic quantum mechanics, based on the general relativity theory proposed by German-born American physicist Albert Einstein (1879–1955). Quantum mechanics are vital in explaining the universe at very small and very large scales, but under typical conditions, Newtonian kinematics are sufficient.

Speed versus Velocity

In everyday conversation, the terms "velocity" and "speed" are often used interchangeably to mean the rate at which an object is moving. In physics, though, these terms have distinct meanings. Speed is the simpler of the two, a ratio of distance traveled to the time spent traveling. It is a scalar quantity, fully described by a simple number. Velocity, however, requires a direction, making it a vector quantity.

The average speed and velocity of an object are easy to calculate using the beginning and end points of its movements and the time of its travel. In the case of an object moving at a uniform speed and uniform direction, the average speed and velocity will accurately describe the object at any instant of its movement. Very few objects move with uniform speed and direction, however.

A football pass, for example, is thrown in an arc. At first it moves forward and upward, but eventually the force of gravity begins to pull it back down again. It slows at the top of its arc and speeds up again as gravity begins working with the down-forward movement of the ball instead of against it. At different moments along the ball’s trajectory, its instantaneous speed might be faster or slower than the average. Its instantaneous velocity might not only be faster or slower, but it may also be in different directions.

Plotting the distance an object has traveled against the time elapsed while it travels is a useful way to visualize changes in an object’s speed over time. (Note that average acceleration is the change in velocity over a period of time, or ā = ∆v / ∆t.) The slope of the line formed by this plot indicates the speed at any given instant.

The key element in distinguishing speed from velocity is displacement. Whereas speed is equal to distance divided by time, velocity is equal to displacement divided by time. Displacement is the absolute distance between the start and end points of an object’s motion. An object moving around a circle, such as the moon in orbit around the earth, is one example. Consider the formula for average velocity, where is average velocity, r is displacement, and t is time taken to complete one orbit.

One orbit takes 27.3 days, or 2,358,720 seconds. Displacement (∆r) is equal to the ending position minus the starting position, which here can be inferred as zero. In a circle, the start and end points of movement are the same. As demonstrated below, this means an object in perfectly circular motion has zero velocity.

Obviously, however, the moon is not sitting still. Its average speed can still be calculated, using the circumference of the circle it travels as a measure of distance (∆d). In this case, plug in the distance traveled by the moon during one full orbit:

The moon’s orbit is about 384,400 kilometers, or 384,400,000 meters. Plugging in these values gives the following:

These rough calculations show that it is possible for an object to be moving quite fast while still having an average velocity of zero.

Similarly, it is easy to imagine a displacement that is smaller than the distance traveled without being zero. A hike to a mountaintop is rarely a straight line—the displacement from the trailhead to the summit may only be a few thousand meters directly north, but hikers can easily travel two or three times that distance on trails that meander back and forth on their way to the top.

Sample Problem

A woman is driving to a new restaurant 6 miles northwest of her home in the city center. The reservations are for 7 p.m. She leaves home at 6 p.m., but along the way, she needs to take several turns and detours. When she arrives, she checks the odometer: she has traveled exactly 30 miles. Looking at the clock reveals the time to be 6:50 p.m. What was the average speed of the vehicle during this trip, in meters per second? What is the average velocity, in meters per second with direction?

Answer:

First, make note of what is known: the trip began at 6 p.m. and ended at 6:50 p.m. Subtracting the end time from the start time, one finds that the trip took 50 minutes, or 3,000 seconds (50 min × 60 s/min).

The odometer measures the distance (∆d) traveled as 30 miles. Convert this to meters.

30 miles × 1,609.347 meters/mile = 48,280.41 meters

For average speed, use the following formula:

Plug in the known values for ∆d and t, and solve:

To calculate average velocity ( ), distance is insufficient. Displacement (∆r) is needed. Displacement measures the absolute distance (in a straight line) between two points and their direction relative to one another. The odometer does not measure displacement, it measures the total distance the car traveled over the road, including all turns and detours. Looking at a map, however, shows that the restaurant is only 6 miles (9,656.06 meters) northwest of the woman’s home. Plugging in this value for displacement yields:

How Kinematics Are Used

More complicated forms of kinematics account for forces like friction and the resistance of the atmosphere; engineers use these sophisticated equations to determine highly precise quantities such as the exact velocity needed for a spacecraft to successfully enter orbit without pulling free of gravity or falling to the surface below. For the average person, being able to comfortably estimate how long it will take to make it across the state in a given time is useful enough.

Bibliography

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Ohanian, Hans C., and John T. Markert. "Motion in Two and Three Dimensions." Physics for Engineers and Scientists. 3rd ed. New York: Norton, 2007. 94–129. Print.

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"Velocity (Mechanics)." Encyclopaedia Britannica. Encyclopaedia Britannica, 10 Dec. 2014. Web. 12 May 2015.

Williams, David R. "Moon Fact Sheet." Lunar and Planetary Science. NASA Goddard Space Flight Center, 25 Apr. 2014. Web. 12 May 2015.