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 ∝ ^{1}_{t} (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 imagine calculating average speed over ever smaller time intervals until we have effectively calculated instantaneous speed. This idea is written symbolically as…
v = 

∆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 distancetime graph, speed corresponds to slope and thus the instantaneous speed of an object with nonconstant 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 
 
instantaneous speed 
 
average velocity 
 
instantaneous velocity 

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 ^{2}_{7} m/s than ^{1}_{4} m/s and 1 mph is more like ^{4}_{9} m/s than ^{1}_{2} 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. 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 reeltoreel 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 phenomena.
 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 is still pretty fast.
 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 × 10^{8} 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 "quasistellar 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.
m/s  km/h  device, event, 

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  
0.5–1.3  1.9–4.6  cockroaches 
1  3.6  nerve impulses, unmyelinated cells 
1  3.6  ocean currents 
1.14  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  43  stadium wave 
14.693  52.894  fastest human: ice skating (Jeremy Wotherspoon) 
12  44  champagne cork 
20  70  rabbits, hares, horses, greyhounds, tuna, sharks 
30  100  typical freeway speed limit 
33  118  cheetahs 
40  140  falling hailstones 
33–83  120–300  hurricane, maximum sustained wind speed 
30–90  105–330  tornado, maximum sustained wind speed 
46.98  169.1  fastest human: baseball pitch (Aroldis Chapman) 
55  200  typical terminal velocity of a 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) 
100  360  nerve impulses, myelinated cells 
113.2  407.5  maximum surface wind gust (Barrow Island, Australia) 
124.22  447.19  fastest streetlegal 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 Oil/Ack Attack) 
180–1200  650–4,400  bullets 
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 (SR71 Blackbird) 
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 
14,000  50,500  New Horizons space probe 
15,000  55,000  Voyager 2 space probe 
17,000  61,000  Voyager 1 space probe 
29,790  107,200  earth in orbit 
190,000  690,000  fastest unmanned spacecraft (Parker Solar Probe) 
220,000  790,000  sun moving through the milky way 
400,000  1,440,000  solar wind near earth 
600,000  2,200,000  milky way through the local super group 
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 