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Speed & 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 before 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 ∝ ⅟t (s constant)
Combining these two rules together gives the definition of speed in symbolic form.
| v = | s | (Note: this is not the final definition.) | |
| t |
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 = | Δs |
| Δt |
The line over the v indicates an average or a mean and the Δ (delta) symbols indicate a change. 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 = | lim | Δs | = | ds |
| Δt → 0 | Δt | 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 this later in this book.
velocity
But Wait, there's more! In order for you or me to calculate the speed of an object we must know how far it's gone and how long it took to get there. Astute observers should then ask a following question …
What do you mean by "how far"? Didn't we learn in the previous section that there are two quantities that can be used to answer the question "how far"?
My but you are wise. Yes indeed, there are two ways to answer that question. When you ask "how far" are you asking for the distance or the displacement? There's a difference between the two quantities and thus a difference between the two answers. To further ruin your life, we're even going to use two different words.
- Speed is the rate of change of distance with time.
- Velocity is the rate of change of displacement with time.
Which means that for the calculus people …
- Speed is the first derivative of distance with respect to time.
- Velocity is the first derivative of displacement with respect to time.
Did I say "ruin your life"? Yes I did, but that's just hyperbole (an intentional exaggeration not meant to be taken literally). I just wanted to get your attention. Velocity and speed mean pretty much the same thing to the average English speaking person, but physics is more precise in its language than is everyday speech.
The situation is not entirely hopeless, however. All the types of speed discussed above also have their counterparts in velocity. Just replace the symbol for distance with the symbol for displacement, et voila. You've got velocity.
|
average speed |
|
instantaneous speed |
||||||||||||||
|
average velocity |
|
instantaneous velocity |
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).
Displacement is measured along the shortest path between two points and thus its magnitude is always less than or equal to the distance. The magnitude of the displacement approaches the 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 examined 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.
| Δt → 0 | ⇒ | v → |v| |
Thus, the instantaneous speed of an object is the magnitude of its instantaneous velocity.
v = |v|
units
Speed and velocity are both measured using the same units. Given that the SI unit of both distance and displacement is the meter and that the SI unit of time is the second, it should be intuitively obvious that the unit of both speed and velocity would be a ratio of two units. The SI unit of speed and velocity is 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 sometimes kph). The United States is an exception in that we use the comparatively archaic 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, everyday units.
| 1 kph = | 1 km | 1000 m | 1 hour | = 0.2777 … m/s ≈ ¼ m/s | ||
| 1 hour | 1 km | 3600 s | ||||
| 1 mph = | 1 mile | 1609 m | 1 hour | = 0.4469 … m/s ≈ ½ m/s | ||
| 1 hour | 1 mile | 3600 s |
The decimal values are accurate to four significant digits, but the fractional values should only be considered rules of thumb (1 mph is really more like 4/10 m/s than ½ 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 the Space Shuttle in knots and its 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 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 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 more compactly 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, phenomena, process | |
|---|---|---|---|
| 10−9 ~ 10−8 | continental plates, fingernail growth, hair 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 | |
| 10.438 | 37.578 | fastest human: running (Usain Bolt) | |
| 12 | 43 | stadium wave | |
| 14.693 | 52.894 | fastest human: ice skating (Jeremy Wotherspoon) | |
| 18 | 64 | champagne cork | |
| 20 | 70 | rabbits, hares, horses, greyhounds, tuna, sharks | |
| 30 | 100 | typical freeway speed limit | |
| 33 | 118 | cheetahs | |
| 36.805 | 132.50 | fastest human: cycling (Sam Whittingham) | |
| 40 | 140 | falling hailstones | |
| 33–83 | 120–300 | hurricane, maximum sustained wind speed | |
| 30–90 | 105–330 | tornado, maximum sustained wind speed | |
| 46.03 | 165.7 | fastest human: baseball pitch (Joel Zumaya) | |
| 55 | 200 | typical terminal velocity of a skydiver | |
| 69.31 | 249.5 | fastest human: tennis serve (Andy Roddick) | |
| 69.833 | 251.400 | fastest human: skiing (Simone Origone) | |
| 80 | 290 | peregrine falcon in a dive | |
| 83 | 295 | very fast golf ball | |
| 100 | 360 | nerve impulses, myelinated cells | |
| 114 | 412 | fastest street-legal car (Ultimate Aero TT SuperCar) | |
| 142.89 | 511.11 | fastest ship (Spirit of Australia) | |
| 148.463 | 534.467 | fastest motorcycle (Fueling Advanced Technologies) | |
| 159.7 | 574.8 | fastest train (Train à Grande Vitesse) | |
| 180–1200 | 650–4,400 | bullets | |
| 200 | 700 | tsunami | |
| 250 | 900 | commercial jet airplane | |
| 274 | 988 | fastest human: sydiving (Joseph Kittinger) | |
| 331 | 1,190 | speed of sound, STP | |
| 340 | 1,225 | speed of sound, sea level | |
| 341.112 | 1,228.02 | fastest experimental car (Thrust SSC) | |
| 343 | 1,235 | speed of sound at room temperature | |
| 980.433 | 3,529.56 | fastest airplane (SR-71 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 | |
| 11,180 | 40,250 | escape velocity | |
| 15,543 | 56,000 | Voyager 2 space probe | |
| 17,100 | 61,600 | Voyager 1 space probe | |
| 29,790 | 107,200 | earth in orbit | |
| 220,000 | 790,000 | sun moving through the milky way | |
| 250,000 | 900,000 | solar wind near earth | |
| 600,000 | 2,200,000 | milky way through the local super group | |
| 124,000,000 | 446,000,000 | speed of light in diamond | |
| 299 792 369 | 1,079,252,530 | protons and antiprotons in the Tevatron, Fermilab | |
| 299,792,458 | 1,079,252,850 | speed of light in vacuum | |
