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 Definition
- Average Speed Formula
- How To Calculate Average Speed
- Average Speed Problems

**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:

$AverageSpeed\left(s\right)=\frac{totaldistance}{totalelapsedtime}$

Average speed contrasts with 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 $55milesperhour\left(mph\right)$. 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 $5.5miles$ in $6minutes$:

$s=\frac{5.5miles}{6mins.}=55mph$

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,

$v=\underset{\u25b3t\to 0}{\mathrm{lim}}\left(\frac{\u25b3x}{\u25b3t}\right)$

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

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.

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:

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

Say you want to find the average speed of a Pacific Bottlenose porpoise. You are told that it can move a distance of $89.7kilometers$ in $3hours$.

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

$s=\frac{89.7km}{3hours}=29.9kilometersperhour\left(kph\right)$

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:

$t=\frac{89.7km}{29.9kph}=3hours$

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

$D=29.9kph\times 3hours=89.7km$

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.

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 $\mathrm{3,600}$ 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).

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

- A tarpon (a type of fish) can travel $105miles$ in $3hours$. What is its average speed?
- A bluefin tuna can swim along and cover $286miles$ in a typical school day of $6.5hours$. What is its average speed while you spend your day in class?
- The world record for fastest speed running backward (while juggling!) is held by Joe Salter, who traveled $\mathrm{5,280}feet$ in $457seconds$. What was his average speed in miles per
*hour*? (multiply by $\mathrm{3,600}$ and then divide by $\mathrm{5,280}$; or multiply by $0.681818$) - A cheetah can cover $0.6miles$ in $36seconds$. What is the cheetah's average speed in miles per
*second*? How about the speed in miles per*hour*? (multiply by $\mathrm{3,600}$) - An orca can travel at an average cruising speed of $8mph$. A great white shark can cruise a distance of $35miles$ in $seven$ hours. What is the great white shark's speed, and which animal cruises faster?
- The fastest human in water covered $22.9meters$ in $10seconds$. The Humbolt squid can travel $399.6meters$ in $60seconds$. 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?

- Calculate the tarpon's average speed like this: $s=\frac{105miles}{3hours}$, which means the fish can travel at an average speed of $35mph$.
- A bluefin tuna's formula would look like this: $s=\frac{286miles}{6.5hours}$, so the fish has an average speed of $44mph$.
- Joe Salter's covered $\mathrm{5,280}feet$ in $457seconds$, so $s=\frac{\mathrm{5,280}feet}{457seconds}$ yields $11.5536feetpersecond$. We multiply this times $\mathrm{3,600}$ (the number of seconds in an hour) and then divide that by $\mathrm{5,280}$ (feet in a mile) to get an average speed of $7.87745mph$.
- The formula for a cheetah's average speed will be $s=\frac{0.6mi.}{36seconds}$, which gives you $0.01666$ (a repeating decimal, so we will approximate with $0.01666$) as miles per
*second*, which you can multiply times $\mathrm{3,600}$ to get an average speed of $60mph$. - An orca can travel at an average cruising speed of $8mph$, while the great white shark's average speed is $s=\frac{35mi.}{7hours}=5mph$. The orca swims faster.
- The fastest human in water swam $22.9meters$ in $10seconds$, so average speed is $s=\frac{22.9m}{10seconds}=2.29meterspersecond$, or $m/s$. The Humbolt squid can travel $399.6meters$ in $60seconds$, so $s=\frac{399.6m}{60seconds}=6.67m/s$, significantly faster than the fastest human swimmer. Let's hope you are never chased by a Humbolt squid!

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.

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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