Speed velocity and acceleration

Average velocity formula

Average velocity formula 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 ∝ 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
t4 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…

 =Δs
Δt

The bar over the v indicates an average or a mean and the Δ (delta) symbols indicate 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 =limΔs =ds
Δt → 0Δtdt

…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

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
speed
 =Δs
Δt
instantaneous
speed
v =limΔs =ds
Δt → 0Δtdt
average
velocity
 =Δr
Δt
instantaneous
velocity
v =limΔr =dr
Δt → 0Δtdt

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.

Δt → 0 ⇒ → ||

The instantaneous speed of an object is 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
ss

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 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 km1,000 m1 hour
1 hour1 km3,600 s
1 kph =0.2777… m/s ≈ ¼ m/s
1 mph =1 mile1,609 m1 hour
1 hour1 mile3,600 s
1 mph =0.4469… m/s ≈ ½ m/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 410 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 1,852 m or 6,076 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 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 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.

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