Speed & Velocity

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
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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.)

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 ∆ (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 in a later section of this chapter.

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 goes and how long it takes 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 different words for the two different concepts.

Which means that for the calculus people …

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.

v =  Δs
Δt
  average
speed		  
v =  lim Δs  =  ds
Δt → 0 Δt dt
  instantaneous
speed
             
v =  Δr
Δt
  average
velocity		  
v =  lim Δr  =  dr
Δt → 0 Δt dt
  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.

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

Summary

Problems

practice

  1. Calculate the size of a light year.

    Astronomical distances are sometimes so large that using meter as the unit is cumbersome. For really large distances the light year is used. A light year is the distance that light would travel in one year in a vacuum. Since the speed of light is fast, and a year is long, the light year is a pretty good unit for astronomy. One light year is about ten trillion meters as the following calculation shows.

         
    v =  Δs  
    Δt
    Δs =  vΔt  = (3.0 × 108 m/s)(365.25 × 24 × 3600 s)
    Δs =  9.46 × 1015 m
       

    Since both the speed of light and the year have exact defined values in the International System of Units, the light year can be stated with an unnecessarily large number of significant digits.

         
    v =  Δs  
    Δt
    Δs =  vΔt  = (299,792,458 m/s)(365.25 × 24 × 3600 s)
    Δs =  9,460,730,472,580,800 m
       

    Some distances in light years are provided below.

    • The distance to Proxima Centauri (the star nearest the sun) is 4.3 light years.
    • The diameter of the Milky Way (a collection of stars that includes the sun and all the stars visible to the naked eye) is about 100,000 light years.
    • The distance to Andromeda (the nearest galaxy outside the Milky Way) is about 2 million light years.
    • The radius of the universe (the observable part of it) is 13.7 ± 0.2 billion light years.
  2. How fast is a point on the equator moving due to the rotation of the earth?

    Notice that no numbers are stated in this problem. When a numerical value is needed to solve a problem and that number is not given, it could mean one of several things.

    • Look it up! It may appear somewhere in the textbook you are using -- on the inside covers, in an appendix, or in the text of the chapter you are currently working on. It may be found in the reference table that some teachers distribute. Standardized exams usually also have their own reference table.
    • Know it! Some numbers are numbers that you should just know. In this problem, there is one relevant number that nearly everyone knows. You may also be expected to memorize certain numbers by an instructor or professor.
    • Calculate it! Maybe there's a way to find the number you need to know using other numbers given in the problem.
    • Forget about it! Maybe you don't really need the number you think you do. Maybe you are on the wrong track. Especially under test conditions, it is highly unlikely that you could be asked a question that requires a numerical value that you can not find, do not know, or can not calculate. Perhaps there is another method to solve this problem.

    In order to calculate speed, you will need distance and time. What distance does a point on the equator move in a convenient period of time? Well, I hope you know that the earth rotates once on its axis every day. You should also know how to calculate the length of a day in seconds. (A day is the period of the earth's rotation, for which an upper case T is the symbol.) During a day, a point on the earth's equator would have traveled a distance equal to the circumference of the earth. The radius of the earth is a number commonly found in textbooks and on reference tables. The problem can now be solved.

                 
    v =  Δs  =  2πr  =  2π(6.4 × 106 m)  = 470 m/s
    Δt T 24 × 3600 s
                 

    That's about one-third greater than the typical speed of sound. An interesting problem to be dealt with later is that if the earth is spinning so rapidly, why then don't things on the equator fly off into space?

  3. I went for a walk one day. I walked north 6.0 km at 6.0 km/h and then west 10 km at 5.0 km/hr. Determine the average speed for the entire journey.

    Solution …

    This problem is deceptively easy. Averaging is taught in elementary school, which makes this an elementary problem. Right?

    6.0 km/h + 5.0 km/h  = 5.5 km/h
    2
    The Wrong Method of Averaging

    Wrong! Wrong! Wrong! Wrong! Wrong! You weren't paying attention in elementary school, were you? This is another example of how memorizing a procedure does not make you smarter (only less ignorant).

    The add-and-divide method of averaging only works when averaging items of equal weight. The average age of the students in a classroom is the sum of their ages divided by the number of students only because each student is considered to have the same weight (a student, is a student, is a student, … ). In this problem, however, the two segments of the walk are significantly different. The second "half" was actually the majority of the walk. It carries more weight than the shorter first "half" of the walk. Thus, the add-and-divide method won't work.

    Let's return to our definition. Since speed is the rate of change of distance with time, we'll need both the distance traveled and the time it took to complete the walk. After we determine both of these numbers, the rest is easy.

    Δt =  Δs     v =  Δs        
    v Δt
    Δt1 =  6.0 km  = 1.0 h v =  6.0 km + 10 km  =  16 km  
    6.0 km/h 1.0 h + 2.0 h 3.0 h
    Δt2 =  10 km  = 2.0 h v =  5.3 km/h  
    5.0 km/h

    Look closely at the calculations on the right side. Notice that the formula contains delta (Δ) symbols and yet I added the distances in the numerator and the times in the denominator. That's because Δ doesn't mean difference, it means change. During the walk my position didn't change from 6.0 km to 10 km, it changed first by 6.0 km and then by 10 km for a total change of 16 km.

  4. A problem for residents of the US only. Convert 60 mph (highway speed) to …
    1. km/h
    2. m/s

    This is an exercise in the factor label method. Write the value as a proper fraction and multiply by ratios equal to one with the intent of canceling the bad units and replacing them with the good ones. Everyone should know (or at least understand) that there are …

    60 × 60 = 3600 seconds

    in an hour. Many Americans who are fans of track and field know that four laps around a 400 m outdoor track is almost one mile.

    1 mile ≈ 4 × 400 m ≈ 1600 meters

    More precisely … actually, most precisely … actually, exactly by definition …

    1 mile = 1609.344 meters

    1. The first answer …
                 
      60 miles   1609.344 m   1 km  ≈ 96.56 km/h
      1 hour 1 mile 1000 m
                 
      For comparison, the speed limit on many of Canada's highways is 100 km/h.
    2. The second answer …
                 
      60 miles   1609.344 m   1 hour  ≈ 26.82 m/s
      1 hour 1 mile 3600 seconds
                 
      You should note that this number is a little bit less than half its value in English units. I find this helpful when trying to interpret answers to problems in an everyday context. I've gotten used to mph from everyday experiences, but I have to be conversant in m/s for my job. A good rule of thumb, therefore is to …
      1. divide by 2 and subtract a little when converting from mph to m/s and to …
      2. multiply by 2 and add a little when converting from m/s to mph.

conceptual

  1. In an unusual move by the New York State Department of Transportation, all of the "speed limit" signs were replaced with "velocity limit" signs.
    1. What would such a sign look like?
    2. How could one travel faster than the old speed limit without violating the new velocity limit?
  2. Which device(s) on a car can be used to change …
    1. its speed?
    2. its velocity but not its speed?
  3. A car driving on a circular test track shows a constant speedometer reading of 100 kph for one lap.
    1. Describe the car's speed during this time.
    2. Describe its velocity.
    3. How do the speed and velocity compare?
  4. Is it possible for an object to have …
    1. constant speed and changing velocity,
    2. changing speed and constant velocity?
  5. Speed is the rate of change of distance with time. Consider a new, as yet, undefined quantity -- the inverse ratio, the rate of change of time with distance.
    1. Under what circumstances would this new quantity
      1. have a large value?
      2. have a small value?
      3. equal zero?
    2. Invent an appropriate name for this new quantity.
  6. Why are the devices in cars called speedometers and not velocitometers?

numerical

  1. The fastest speed achieved by a snail in the Guinness Gastropod Championship, held over a 330 mm (13 in) course in the O'Conor Don pub in central London is held by a mollusk called Archie, which took 2 minutes and 20 seconds to cover the course. Determine Archie's speed in m/s and km/h.
  2. A moving driver not anticipating an accident can apply the brakes fully in about 0.5 s. How far would a car driving down the freeway at 30 m/s travel in this time?
  3. In an experiment at James Cook University in Australia, a researcher put the larvae of tropical fish in a special tank to measure their swimming speeds. The tank generates an adjustable current that the fish must swim against. The most proficient swimmer was a surgeonfish larva that maintained a 13.5 cm/s swim for an equivalent distance of 94 km without a rest. For how long was the larva swimming?
  4. A high speed video camera running at 180 frames per second was used to record a player kicking a soccer ball. Each square on the grid behind the ball is 10 cm on a side.
    1. View the video and then determine the speed of the soccer ball. The video is available in .gif or .mov formats.
    2. Penalty kicks in soccer take place 11.0 m away from the goal. Calculate the time it takes the ball to cover this distance.
  5. When designing aircraft it is common to place them in a wind tunnel: a closed room where air is blown at high speed. As an option, some tests can be performed in an indoor hyperballistic range. In one such range, aircraft models are projected at 9000 m/s (20,000 mph) into a catching device designed to recover them intact. Ultra-high-speed cameras with laser illumination then photograph the model at exposures of 20 ns. How far will such a model move while it is being photographed?
  6. It takes a plane flying at 150 km/h 3.0 minutes to circle a cloud at an altitude of 3,000 m. What is the diameter of the cloud?
  7. The three-toed sloth is the slowest land mammal. On the ground, the sloth moves at an average speed of 0.23 m/s (0.5 mph). The cheetah is the fastest land mammal. A cheetah is capable of speeds up to 31 m/s (70 mph) for brief periods. If a cheetah were to run at top speed for 3.0 s, how long would it take the sloth to catch up?
  8. Calculate the orbital speed of the moon.
  9. Calculate the orbital speed of the earth.
  10. Calculate the size of a …
    1. light-day
    2. light-hour
    3. light-minute
    4. light-second
    5. light-millisecond
    6. light-microsecond
    7. light-nanosecond
    8. light-picosecond
    9. light-femtosecond
  11. Radio waves travel at the speed of light. Calculate the "round trip light time" for the following astronomical objects. That is, how long would it take a radio signal to travel from the earth to the object and back?
    1. earth's moon
    2. the sun
    3. Mars (when closest to the earth)
    4. Pluto (when farthest from the earth)
  12. At one time, the great goal of middle distance runners was the four minute mile.
    1. What is the average speed of a runner capable of this feat in mph and m/s?
    2. How long would it take to complete a marathon (26 miles 385 yards) at this pace?
  13. Isaac Newton was born in Lincolnshire, near Grantham, on 25 December 1642, and died in Kensington, London, on 20 March 1727. Grantham is approximately 160 km north of London.
    1. Calculate the average velocity of Mr. Newton over his lifetime in m/s.
    2. Why does this problem ask for the average velocity and not the instantaneous velocity?
    3. Why does this problem ask for the average velocity and not the average speed?
    4. If Mr. Newton had instead lived until 13 June 1811, what total displacement would he have experienced over his lifetime? Where would he have died?
  14. A stunt crew is planning a chase scene for a movie. The script calls for a car to drive across a railroad track moments before a train enters the crossing. (Warning: Don't try this at home!) The locomotive engineer recommends a speed of 10 m/s for safety and the director wants the car moving at 30 m/s for excitement. Where should the rear of the car be when the train is at the following distances from the crossing …
    1. 20 m,
    2. 10 m,
    3. 5 m, and
    4. 1 m?
  15. The same crew members from the previous problem, now have to prepare another stunt for the same movie. They plan to have a second car drive off a ramp at the train. The jump will be timed so that an empty flatcar will roll into the crossing and the pursuing car will then be able to slip through the gap and continue the chase. (Warning: Don't try this at home! If you do, you are seriously stupid.) The flatcar is 16 m long by 3 m wide and that the pursuing car is 4 m long by 2 m wide. The train is still moving at 10 m/s. Determine the minimum speed at which the car must be driven off the ramp.
  16. Here are some data for a three segment trip to an exotic distant location. Calculate the missing data and complete the table below.

    A Three Segment Trip
    trip segment distance traveled elapsed time average speed
    by plane 6930 km ? 965 km/h
    by taxi 201 km 2.90 h ?
    on foot ? 5.75 h 4.50 km/h
    entire trip ? ? ?

statistical

  1. hawaiian-chain.txt
    The Hawaiian Island chain is more than just the visible islands. It also includes a few dozen seamounts -- islands that have eroded down below sea level. The combined Hawaiian Islands--Emperor Seamounts chain is a series of volcanic structures formed by a single, long-lived plume of magma referred to as a "hotspot". The hotspot stayed fixed as the pacific plate slowly moved over it, resulting in a chain of volcanoes stretching from the Aleutian Islands off the coast of Alaska to Mount Kilauea on the Big Island of Hawaii in the tropics. Use the data set in the accompanying text file to determine the speed of the Pacific plate in cm/yr.
    The columns in this data set are as follows:
    1. volcano number
    2. volcano name
    3. volcano age (millions of years)
    4. distance from Kilauea (km)
    5. error in age (millions of years)
    6. error in distance (km)
  2. track-events.txt
    This file gives the world record times for eight track events as of August 1999. Calculate the average speed of each record holder. From these numbers determine …
    1. the effect of gender on speed
    2. the effect of distance on speed
    in track events at the elite level.
    The columns in this data set are as follows:
    1. event distance (m)
    2. men's record time, hours portion
    3. men's record time, minutes portion
    4. men's record time, seconds portion
    5. women's record time, hours portion
    6. women's record time, minutes portion
    7. women's record time, seconds portion

investigative

  1. Repeat the last statistical problem above, but this time …
    1. use the world records in swimming,
    2. use the results from a local track and field competition,
    3. use the results from a local swimming competition, or
    4. use the results from several different years to determine the trend in the speed of women compared to the speed of men. Predict the year when they will equal.
  2. The twin spacecraft Voyager 1 and Voyager 2 were launched by NASA in the summer of 1977 from Cape Canaveral, Florida. As originally designed, the Voyagers were to conduct close up studies of Jupiter and Saturn. Eventually, Voyager 2 would go on to explore Uranus and Neptune. The spacecraft are still operating and continue to return data about interplanetary space. Range, velocity, and round trip light time for the Voyagers are available at the Voyager Project web site. Using the data at this site, determine the following quantities in m/s …
    1. the instantaneous and average speed of each spacecraft,
    2. the magnitude of the instantaneous and average velocity of each spacecraft, and
    3. the speed of light.
  3. Obtain the necessary biographical information needed to determine the magnitude of the average velocity of a dead physicist over his or her lifetime in m/s. For a list of physicist with online biographies see Yahoo! Science: Physics: Physicists.
  4. Obtain an airline timetable for the planes departing from a hub airport. Find a flight that continues on to a second destination after a brief layover. Use the data to calculate the average speed of this plane …
    1. from the hub to the primary destination,
    2. from the primary destination to the secondary destination, and
    3. from the hub to the secondary destination.
  5. Obtain the door-to-door travel info from your home to the center of another city on a nonstop flight. Include the duration of the two taxi rides, arrival and departure times for the plane, distance of each taxi ride, distance of the plane flight, recommended check in time, and an estimate of the time it takes to exit the plane, gather up your luggage, and hail a taxi. Calculate the average speed of …
    1. the first taxi ride,
    2. the plane flight,
    3. the second taxi ride, and
    4. the entire trip.
  6. Assuming it were possible, how long would it take to travel from the earth to Mars along a straight line on the day of their closest approach by …
    1. walking at a casual pace?
    2. running at marathon speeds?
    3. driving at freeways speeds?
    4. flying in a commercial airplane?
    5. riding a rifle bullet?
    6. riding a beam of light?

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