## Average Speed Definition

Average speed combines two ideas in two words: average, meaning a mean derived from a lot of individual data points, and speed, which is a change in position.

You can calculate the average speed for any type of motion if you can time the motion and measure the distance.

## Average Speed Formula

Average speed is the total distance traveled for the object in question divided by the total elapsed time taken to travel the distance, the total period of time. The average speed formula is:

Average speed contrasts with instantaneous speed.

### Instantaneous Speed

Average speed takes into account the entirety of an event, such as a car accelerating from a stop, speeding up, traveling for a while, then slowing at a yellow light, and finally stopping.

A car travels at different speeds. At any given moment, the car is not traveling at . It might be 0 mph, then 7 mph in another instant, then 53 mph, then 61 mph, and finally 3 mph before returning to 0 mph.

To simplify measurements and make progress on a physics or mathematics problem, you take the average speed of all the discrete events, saying the car traveled in :

All those other measurements, at particular moments in the journey, are instantaneous speeds. In most cases you do not need to know the formula for instantaneous speed, v, finding the limit as the change in time (the “instant”) approaches 0:

Instantaneous speeds fluctuate throughout a timed event. Finding the average speed is far easier – and usually far more useful – than calculating instantaneous speed.

### Scalar and Vector Quantities

Speed is a scalar quantity. It does not have a direction. It only has a size, meaning a magnitude or scale. Scalar quantities can go from $0$ (no speed) to infinitely fast.

A vector quantity has size and direction, as with the movement of an airplane in the sky. Velocity is a vector quantity.

Speed, being a scalar quantity, can never be less than $0$. Average and instantaneous speeds are always scalar quantities, which means you can always measure them with a number. Distance and time are also scalar quantities and can also be measured with numbers.

## How To Calculate Average Speed

To calculate the average speed of an object, you must know the total distance an object travels and the total elapsed time of its whole journey.

The distance/speed/time triangle is handy for calculating this and two other scalar quantities (distance and time):

The three parts in the triangle are set up in their correct positions mathematically:

1. To get average speed, $s$, divide total distance by elapsed time: $\frac{D}{t}$
2. To get elapsed time, $t$, divide total distance by speed: $\frac{D}{s}$
3. To get distance, $D$, multiply speed times the amount of time:

Say you want to find the average speed of a Pacific Bottlenose porpoise. You are told that it can move a distance of in .

Plug those two given numbers into the triangle in their two corners to get:

If you know two of the three variables, distance, time, and speed, then you can use algebra to find what you're missing.

If you need the total time, you must have the distance and speed. You plug those two scalar quantities into their parts of the triangle to get:

If you need the total distance, you must have the speed and the time:

Average speed is especially useful because it takes into account the reality of an event, rather than assuming something or someone is moving at a constant speed.

The porpoise could have started slowly, sped up, paused to play, and continued. That three-toed sloth may have stopped for a moment to catch its breath before hurrying onward. You might have to make numerous stops when walking a dog, but in all three cases, you can easily calculate average speed by dividing the total distance traveled by total elapsed time.

### A Cautionary Note

Average speed is often derived from units of distance or time that must convert to other units for the final answer. Use care when doing this. Common conversions are to multiply units per second by $60$ or $3,600$ to get units per minute and units per hour. Just make sure your answer is given in the correct unit of time.

If only one unit is being changed, you will have only one mathematical operation to perform (multiply seconds to get minutes or hours, for example). If two units are changed (feet per second to miles per hour), you have to both multiply and divide (or multiply by a decimal value).

## Average Speed Problems

Test what you've learned with the example word problems:

1. A tarpon (a type of fish) can travel in . What is its average speed?
2. A bluefin tuna can swim along and cover in a typical school day of . What is its average speed while you spend your day in class?
3. The world record for fastest speed running backward (while juggling!) is held by Joe Salter, who traveled in . What was his average speed in miles per hour? (multiply by $3,600$ and then divide by $5,280$; or multiply by $0.681818$)
4. A cheetah can cover in . What is the cheetah's average speed in miles per second? How about the speed in miles per hour? (multiply by $3,600$)
5. An orca can travel at an average cruising speed of . A great white shark can cruise a distance of in $seven$ hours. What is the great white shark's speed, and which animal cruises faster?
6. The fastest human in water covered in . The Humbolt squid can travel in . You need to calculate the average speed of the fastest human and a Humbolt squid to know who can outrace whom.

We know you will do the work first, before checking these answers, right?

1. Calculate the tarpon's average speed like this: , which means the fish can travel at an average speed of .
2. A bluefin tuna's formula would look like this: , so the fish has an average speed of .
3. Joe Salter's covered in , so yields . We multiply this times $3,600$ (the number of seconds in an hour) and then divide that by $5,280$ (feet in a mile) to get an average speed of .
4. The formula for a cheetah's average speed will be , which gives you $0.01666$ (a repeating decimal, so we will approximate with $0.01666$) as miles per second, which you can multiply times $3,600$ to get an average speed of .
5. An orca can travel at an average cruising speed of , while the great white shark's average speed is . The orca swims faster.
6. The fastest human in water swam in , so average speed is , or $m/s$. The Humbolt squid can travel in , so , significantly faster than the fastest human swimmer. Let's hope you are never chased by a Humbolt squid!

## What you learned:

After working your way through this lesson and video, you should have learned:

• The defition of average speed
• The difference between scalar and vector quantities
• The average speed formula
• How to find or calculate average speed
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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