The Physics
Hypertextbook
Opus in profectus

# Speed and Velocity

## Discussion

### speed

What's the difference between two identical objects traveling at different speeds? Nearly everyone knows that the one moving faster (the one with the greater speed) will go farther than the one moving slower in the same amount of time. Either that or they'll tell you that the one moving faster will get where it's going sooner than the slower one. Whatever speed is, it involves both distance and time. "Faster" means either "farther" (greater distance) or "sooner" (less time).

Doubling one's speed would mean doubling one's distance traveled in a given amount of time. Doubling one's speed would also mean halving the time required to travel a given distance. If you know a little about mathematics, these statements are meaningful and useful. (The symbol v is used for speed because of the association between speed and velocity, which will be discussed shortly.)

• Speed is directly proportional to distance when time is constant: v ∝ s (t constant)
• Speed is inversely proportional to time when distance is constant: v ∝ 1t (s constant)

Combining these two rules together gives the definition of speed in symbolic form.

 v = s t
 ☞ This is not the final definition.

Don't like symbols? Well then, here's another way to define speed. Speed is the rate of change of distance with time.

In order to calculate the speed of an object we must know how far it's gone and how long it took to get there. "Farther" and "sooner" correspond to "faster". Let's say you drove a car from New York to Boston. The distance by road is roughly 300 km (200 miles). If the trip takes four hours, what was your speed? Applying the formula above gives…

 v = s ≈ 300 km = 75 km/h t 4 hour

This is the answer the equation gives us, but how right is it? Was 75 kph the speed of the car? Yes, of course it was… Well, maybe, I guess… No, it couldn't have been the speed. Unless you live in a world where cars have some kind of exceptional cruise control and traffic flows in some ideal manner, your speed during this hypothetical journey must certainly have varied. Thus, the number calculated above is not the speed of the car, it's the average speed for the entire journey. In order to emphasize this point, the equation is sometimes modified as follows…

 v = ∆s ∆t

The bar over the v indicates an average or a mean and the (delta) symbol indicates a change. Read it as "vee bar is delta vee over delta tee". This is the quantity we calculated for our hypothetical trip.

In contrast, a car's speedometer shows its instantaneous speed, that is, the speed determined over a very small interval of time — an instant. Ideally this interval should be as close to zero as possible, but in reality we are limited by the sensitivity of our measuring devices. Mentally, however, it is possible to imagine calculating average speed over ever smaller time intervals until we have effectively calculated instantaneous speed. This idea is written symbolically as…

v =
 lim ∆t→0
s  =  ds
t dt

or, in the language of calculus speed is the first derivative of distance with respect to time.

If you haven't dealt with calculus, don't sweat this definition too much. There are other, simpler ways to find the instantaneous speed of a moving object. On a distance-time graph, speed corresponds to slope and thus the instantaneous speed of an object with non-constant speed can be found from the slope of a line tangent to its curve. We'll deal with that later in this book.

### velocity

In order to calculate the speed of an object we need to know how far it's gone and how long it took to get there. A wise person would then ask…

What do you mean by how far? Do you want the distance or the displacement?

A wise person, once upon a time

Your choice of answer to this question determines what you calculate — speed or velocity.

• Average speed is the rate of change of distance with time.
• Average velocity is the rate of change of displacement with time.

And for the calculus people out there…

• Instantaneous speed is the first derivative of distance with respect to time.
• Instantaneous velocity is the first derivative of displacement with respect to time.

Speed and velocity are related in much the same way that distance and displacement are related. Speed is a scalar and velocity is a vector. Speed gets the symbol v (italic) and velocity gets the symbol v (boldface). Average values get a bar over the symbol.

average
speed

 v = ∆s ∆t
instantaneous
speed

v =
 lim ∆t→0
s = ds
tdt
average
velocity

 v = ∆s ∆t
instantaneous
velocity

v =
 lim ∆t→0
s = ds
tdt

Displacement is measured along the shortest path between two points and its magnitude is always less than or equal to the distance. The magnitude of displacement approaches distance as distance approaches zero. That is, distance and displacement are effectively the same (have the same magnitude) when the interval examined is "small". Since speed is based on distance and velocity is based on displacement, these two quantities are effectively the same (have the same magnitude) when the time interval is "small" or, in the language of calculus, the magnitude of an object's average velocity approaches its average speed as the time interval approaches zero.

 ∆t → 0 ⇒ v → |v|

The instantaneous speed of an object is then the magnitude of its instantaneous velocity.

v = |v|

Speed tells you how fast. Velocity tells you how fast and in what direction.

### units

Speed and velocity are both measured using the same units. The SI unit of distance and displacement is the meter. The SI unit of time is the second. The SI unit of speed and velocity is the ratio of two — the meter per second.

 ⎡⎢⎣ m = m ⎤⎥⎦ s s

This unit is only rarely used outside scientific and academic circles. Most people on this planet measure speeds in kilometer per hour (km/h or kph). The United States is an exception in that we use the older mile per hour (mi/h or mph). Let's determine the conversion factors so that we can relate speeds measured in m/s with the more familiar units.

 1 kph = 1 km 1000 m 1 hour 1 hour 1 km 3600 s 1 kph = 0.2777… m/s ≈ ¼ m/s 1 mph = 1 mile 1609 m 1 hour 1 hour 1 mile 3600 s 1 mph = 0.4469… m/s ≈ ½ m/s

The decimal values shown above are accurate to four significant digits, but the fractional values should only be considered rules of thumb (1 kph is really more like 27 m/s than 14 m/s and 1 mph is more like 49 m/s than 12 m/s).

The ratio of any unit of distance to any unit of time is a unit of speed.

• The speeds of ships, planes, and rockets are often stated in knots. One knot is one nautical mile per hour — a nautical mile being 1852 m or 6076 feet and an hour being 3600 s. NASA still reports the speed of its rockets in knots and their downrange distance in nautical miles. One knot is approximately 0.5144 m/s.
• The slowest speeds are measured over the longest time periods. The continental plates creep across the surface of the Earth at the geologically slow rate of 1–10 cm/year or 1–10 m/century — about the same speed that fingernails and hair grow.
• Audio cassette tape travels at 1⅞ inches per second (ips). When magnetic tape was first invented, it was spooled on to open reels like movie film. These early reel-to-reel tape recorders ran the tape through at 15 ips. Later models could also record at half this speed (7½ ips) and then half of that (3¾ ips) and then some at half of that (1⅞ ips). When the audio cassette standard was being formulated, it was decided that the last of these values would be sufficient for the new medium. One inch per second is exactly 0.0254 m/s by definition.

Sometimes, the speed of an object is described relative to the speed of something else; preferably some physical phenomenon.

• Aerodynamics is the study of moving air and how objects interact with it. In this field, the speed of an object is often measured relative to the speed of sound. This ratio is known as the Mach number. The speed of sound is roughly 295 m/s (660 mph) at the altitude at which commercial jet aircraft normally fly. The now decommissioned British Airways and Air France supersonic Concorde cruised at 600 m/s (1340 mph). Simple division shows that this speed is roughly twice the speed of sound or Mach 2.0, which is exceptionally fast. A Boeing 777, in comparison, cruises at 248 m/s (555 mph) or Mach 0.8, which only seems slow in comparison to the Concorde.
• The speed of light in a vacuum is defined in the SI system to be 299,792,458 m/s (about a billion km/h). This is usually stated with a more reasonable precision as 3.00 × 108 m/s. The speed of light in a vacuum is assigned the symbol c (italic) when used in an equation and c (roman) when used as a unit. The speed of light in a vacuum is a universal limit, so real objects always move slower than c. It is used frequently in particle physics and the astronomy of distant objects. The most distant observed objects are quasars; short for "quasi-stellar radio objects". They are visually similar to stars (the prefix quasi means resembling) but emit far more energy than any star possibly could. They lie at the edges of the observable universe and are rushing away from us at incredible speeds. The most distant quasars are moving away from us at nearly 0.9 c. By the way, the symbol c was chosen not because the speed of light is a universal constant (which it is) but because it is the first letter of the Latin word for swiftness — celeritas.
Selected speeds (slowest to fastest)
m/s km/h device, event, phenomenon, process
10−9~10−8 continental plates, hair growth, fingernail growth
10−4 human sperm cells
10−3 snails
0.013 0.045 ketchup pouring from a bottle
10−1 sloths, tortoises, turtles
0.65–1.29 2.34–4.64 cockroaches
1 3.6 nerve impulses, unmyelinated cells
1 3.6 ocean currents
0.06–1.14 0.22–4.10 manatees
1.3 4.8 human, typical walking pace
2.391 8.608 fastest human: swimming (César Cielo)
8 30 maximum comfortable elevator speed
10 40 dolphins, porpoises, whales
10 40 falling raindrops
10.422 37.520 fastest human: running (Usain Bolt)
12 44 champagne cork
15.223 54.803 fastest human: ice skating (Pavel Kulizhnikov)
20 70 rabbits, hares, horses, greyhounds, tuna, sharks
30 100 typical freeway speed limit
33 118 cheetahs
34.42 123.9 fastest human: softball pitch (Monica Abbott)
40 140 falling hailstones
42.47 152.9 fastest human: flying disc throw (Simon Lizotte)
46.98 169.1 fastest human: baseball pitch (Aroldis Chapman)
55 200 terminal velocity of a typical skydiver
70.8217 254.958 fastest human: skiing (Ivan Origone)
73.06 263 fastest human: tennis serve (Sam Groth)
80 290 peregrine falcon in a dive
82 295 very fast golf ball
82.211 296.00 fastest human: cycling (Denise Korenek Mueller)
33–83 120–300 hurricane, maximum sustained wind speed
30–90 105–330 tornado, maximum sustained wind speed
100 360 nerve impulses, myelinated cells
113.2 407.5 maximum surface wind gust (Barrow Island, Australia)
124.22 447.19 fastest street-legal car (Koenigsegg Agera RS)
142.89 511.11 fastest ship (Spirit of Australia)
159.7 574.8 fastest train (Train à Grande Vitesse)
168.249 605.697 fastest motorcycle (Top 1 Ack Attack)
200 700 tsunami
250 900 commercial jet airplane
331 1,190 speed of sound in air, STP
340 1,225 speed of sound in air, sea level
341.4031 1,229.051 fastest experimental car (Thrust SSC)
343 1,235 speed of sound in air, room temperature
377.1 1,357.6 fastest human: skydiving (Felix Baumgartner)
980.433 3,529.56 fastest airplane (SR-71 Blackbird)
180–1,200 650–4,400 bullets
1,500 5,400 speed of sound in water
2,000 6,000 seismic waves
6,900 25,000 detonation velocity of TNT
8,000 29,000 space shuttle in orbit
11,094 39,938 fastest manned spacecraft (Apollo 10)
11,180 40,250 escape velocity on the surface of the Earth
13,900 50,400 New Horizons space probe
15,400 55,400 Voyager 2 space probe
17,000 61,200 Voyager 1 space probe
29,790 107,200 Earth in orbit
190,000 690,000 fastest unmanned spacecraft (Parker Solar Probe)
248,000 892,000 Sun moving through the Milky Way
300,000 1,100,000 solar wind near earth
370,000 1,330,000 Milky Way through the cosmic microwave background
60,000,000 216,000,000 Project Starshot, proposed interstellar space probe
124,000,000 446,000,000 speed of light in diamond
225,000,000 810,000,000 speed of light in water
299,792,369 1,079,252,530 protons and antiprotons in the Tevatron, Fermilab
299,792,455 1,079,252,840 protons in the Large Hadron Collider, CERN
299,792,458 1,079,252,850 speed of light in a vacuum