Speed & Velocity
   

Speed is defined as distance, divided by the time needed to cover that distance. Actually, because the speed can change from moment to moment, this is actually the definition of average speed. We're going to use the variable 'd' for distance, 't' for the time, and 'v' for the speed. We're going to put a bar over the v to signify that it is an average: . So, written in equation form, our definition of average speed is

Another way of saying this is that the speed is the "rate of change of position". The change of position is the distance traveled. The word rate means "divided by time". The word per means 'divided by', so yet another way to say this is that average speed is "distance per time".

If we solve this for d, we get

 

  

Which skier will reach the finish line first, assuming they start simultaneously?

 

  

Examples  The speed of sound is about 340 m/sec.

(a) How far will sound travel in one minute?

      First, since we are trying to calculate distance, use , not .

                     

(b) How much time does it take sound to travel one mile (1609 meters)?

     Here, we are trying to calculate time, not speed or distance. We need, therefore, to solve the equation for t.

Start with .

Divide both sides of the equation by . This yields us

Now we can plug in numbers:

               

This is handy to know if you are ever out hiking and a thunderstorm pops up. Every 5-second delay between when you see the lightning flash and when you hear the thunder means one mile of distance between you and the lightning.

 

1. How far will sound travel in one millisecond? SOLUTION VIDEO

2. How much time does it take sound to travel one kilometer? SOLUTION VIDEO

3. How much time does it take sound to travel from me to you, assuming we are 2 meters apart?

4. Using a yardstick, meterstick, or tape measure, lay out a known distance. Make it long, like the length of your house or apartment building. Time how long it takes you to (a) walk and (b) run that distance. Calculate the two average speeds. How many times faster is your running speed than walking?

Why are we using v to stand for speed? Why not s? The 'v' comes from the word velocity. Velocity is a so-called vector. We'll be talking about vectors in greater detail later, but for now, all you need to know is that vectors have a magnitude and a direction. Speed is the magnitude of the velocity. This may make more sense with a concrete example. If you say a car is traveling at 30 miles per hour, you have described its speed. If you say it's moving to the southwest, you've described the direction of its motion. If you say both speed and direction, you have described its velocity. In practice, we will often use the words 'speed' and 'velocity' interchangably, but the distinction will be important when we are talking about motion in two or three dimensions, later.

 

  
Instantaneous vs. Average Speed As mentioned above, we have been talking here about average speed. If you travel in a car 400 kilometers in 4 hours, your average speed was 100 km/hr. However, in the real world, you would not travel at exactly 100 km/hr for that entire 4-hour period. You would have to slow down in heavy traffic and stop for red lights, for example, and at other times you would be traveling faster than the average speed. The speed you are traveling at any moment (or instant) of time is called the instantaneous speed.

5. Does the speedometer in a car measure average or instantaneous speed?

 

 

Additional Activities and Practice

6. After school on May 10, 2010, a sixteen-year-old girl from Asheville died tragically when her car veered into oncoming traffic. Investigators determined that she was texting on her cell phone as she was driving. Drivers who text typically take their eyes off the road for about 5 seconds each "glance" at the phone. Accident investigators determined the Asheville tenth-grader was traveling about 52 miles per hour when she plowed into an oncoming pickup truck. How far did her car travel while she was looking at her phone?

7. Human nerve signals travel at about 270 miles/hour. How much time does it take a signal to get from your brain to the muscles of the hand, if that distance is 0.85 meters?

8. In 1946, an earthquake in the ocean near the Aleutian Islands, Alaska created a tsunami that hit Hawaii 4 and a half hours later, killing about 160 people. The distance from the earthquake epicenter to the first shoreline in Hawaii was 3600 km. How fast did the tsunami travel, in kilometers per hour?

9. The North American plate and the Eurasian plate of the Earth’s crust were once touching, but gradually separated from each other by the continuous creation of new ocean-floor crust at the Mid-Atlantic Ridge. They are currently observed to be separating at an average rate of 25 mm/year.

(a) What is that speed in miles per hour? (Use the FL method.)
(b) How many years will it take for the two plates to get 1.0 kilometer farther apart than they are now?

10. The world’s fastest jet-powered airplane was the now-retired SR-71 Blackbird, which could travel at 3370 kilometers per hour. These airplanes were used during the Cold War as a spy plane against the Soviet Union. How much time would it take an SR-71 to fly from Greensboro to Paris, France, a distance of 6550 km? Show your work.

11. The Cape Hatteras Lighthouse was built in 1870 on Hatteras Island, NC. It was constructed 1500 feet from the ocean. Barrier islands such as Hatteras are essentially sand bars sticking out above the water, and erosion and redeposition of sand causes these islands to gradually move. By 1999, the Cape Hatteras Lighthouse was about 100 feet from the water’s edge and in danger of toppling. In June and July 1999, the 4800-ton lighthouse was moved 2900 feet over the course of 23 days, placing it about 1600 feet from the ocean. (NOTE: 1 m = 3.28 ft)(Information source: “Moving the Cape Hatteras Light” by Anne Douglas, located at http://www.essortment.com/all/capehatterasli_rnhu.htm.)

(a) What was the average speed of the lighthouse during the 23 days it was being moved? Express your answer in feet/day.
(b)  Convert your answer to part (a) into m/day using the FL method.
(c) What was the average speed of recession of the shoreline, from 1870 to 1999? Express your answer in feet/year.
(d) Convert your answer to part (c) to m/day using the FL method.

12. A "light-year" is a unit of distance, equal to the distance light travels in a year. The speed of light is 3.00x108 m/sec. Calculate the length of a light-year, in meters.

13. The Sun is 150 million kilometers away from Earth. If you were to drive a car at highway speeds (88 km/hr), how much time would it take to get there? Express your answer in the most evocative, easy-to-understand unit.

  
   

 

 

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