DESCRIPTION OF INTERACTIVE

Distance

In the video, the robots make use of bar codes on the floor to know how far they have to travel in each direction in order to fulfil an order. Each BettyBot can travel up to 19.2 kilometres (12 miles) a day.

That’s quite the distance!

Distance is how far an object has travelled, typically measured in metres, or multiples of metres such as kilometres (km) or centimetres (cm). It is given the symbol “d.”

Here is a floor plan of the Quiet Logistics warehouse: alt tag: This is an image of the floor plan the robots use to get around the warehouse.

Let’s assume that each grid square represents a 1m by 1m square on the floor of the warehouse. If BettyBot travelled directly from point A to point B as shown in the diagram then BettyBot would have travelled a distance of 34 m. If a BettyBot robot only travelled back and forth in a straight line all day from point A to point B, could we figure out how many passes it would have to make to travel 19.2 km in total?

Note: The Metric System. 

In Canada, as with most countries around the world, we measure using the metric system. Some base units in this system are the kilogram (for mass), the metre (for length), and the second (for time). We will be using these base units throughout this course.

However, when measuring things that are really large (the distance to the nearest galaxy), or really small (the width of an atom), it might not be appropriate to state our answer in metres. This is where various metric number prefixes come in.

You are likely already familiar with some of these prefixes:

While we won’t be using ALL of the following prefixes in this course (some of them are pretty obscure), you may find it useful to keep chart handy.

So, for instance, if we measure 56 km, we know we have 56 m x 1 000 = 56 000 m (the “k” tells us we’re using the kilo- prefix, so we multiply our original number by the multiplier of 1 000).

If we have 740 “n”s, we know we have 740 s x 0.000 000 001 = 0.000 000 740 s (the “n” tells us we’re using the nano- prefix, so we multiply our original number by the multiplier 0.000 000 001).

Try the following questions to see if you’ve got the hang of it:

1. How many grams are 14 Teragrams (Tg)?

  1. 14 000 g

  2. 14 000 000 000 000 g

  3. 14 000 000 000 g

  4. 0.000 000 000 014 g

ANSWER: b. 14 000 000 000 000 g

14Tg = 14 g x 1 000 000 000 000

         = 14 000 000 000 000 g

 

2. How many seconds are 182 deciseconds (ds)?

  1. 0.182 s

  2. 1.82 s

  3. 18.2 s

  4. 1 820 s

ANSWER: c. 18.2 s

182 ds = 182 s x 0.1

            = 18.2 s

 

3. How many metres are 9.3 megametres (Mm)?

  1. 9 300 000 m

  2. 9 300 000 000 m

  3. 0.009 3 m

  4. 0.000 009 3 m

ANSWER: a. 9 300 000 m

9.3 Mm = 9.3 m x 1,000,000

             = 9 300 000 m

 

Note that capitalization of the prefix matters. A megametre (Mm) is very different than a millimetre (mm).

A measurement of 19.2 km is the same as 19.2 m x 1 000 = 19 200 m. To figure out how many passes between points A and B the robot would have to complete, divide 19 200 m by 34 m. This gives us an answer of 564.7 passes. A lot of travel for a little robot!

Position

Knowing how far each robot has travelled would be useful for monitoring when moving parts need to be replaced, or when a particular robot should return to a charging station, but it is not a useful measurement for deciding which robot is the best choice for making a pick up.

For that, we use position. Position is the location of an object with respect to a reference point (or origin). Position can be determined by measuring a straight line path from the reference point (or origin) to the location of the object, including the direction of the path to the object. Like distance, position is typically measured in metres, or multiples of metres such as kilometres (km) or centimetres (cm). It is given the symbol,  d → {"version":"1.1","math":" d → "} . That little arrow above the “d” indicates that in addition to providing the distance and unit, we also need to provide a direction, such as “east,” “north,” “above,” “behind,” etc.

Vector vs. Scalar

In Physics, we use two types of measurements: scalars and vectors. 

Scalars are quantities that do not require a direction. Examples include mass (m), temperature (T), volume (V), and distance (d).

Vectors are quantities that do require a direction, like position (\vec{d}). The arrow in the quantity’s symbol reminds us to include the direction in the final answer. Later in the course, we will also see vector quantities of velocity (\vec{v}), acceleration (\vec{a}), and force (\vec{F}), all of which will be accompanied by a direction.

The proper notation for indicating direction is to list it after the unit of the measurement in question, in [square brackets]. For instance, an object’s position might be stated as \vec{d}=5m[south] or \vec{d}=78cm[left].

Displacement

As each BettyBot moves around the warehouse floor, the central computer must keep track of each robot’s change in position.

Displacement is the change in position of an object, includes the direction of the change in position. It is given the symbol,  △ d → \Delta \vec{d}, and is measured in metres, just like distance and position. The symbol can be broken down into two parts: you might recognize the symbol for position (\vec{d}), where the vector notation (the small arrow above the symbol) reminds us that we must include a direction. The triangle is actually the Greek letter Delta (\Delta), which represents “the change in.”

Delta is a shorthand that is often used in Physics whenever we want to demonstrate how something is changing. For example, if it warmed up outside from 10 ˚C to 15 ˚C, we would write \Delta T=5^{o}C, because the change in temperature is 5 ˚C.

Mathematically, we can write:

\Delta T=T_{f}-T_{i}

\Delta T=15^{o}C-10^{o}C

\Delta T=5^{o}C

Any time we want to find the change in something, we can find the difference between two quantities, which involves subtraction.

Significant Figures

When completing computations in Physics, we want to make sure our solutions are as precise as our initial measurements.

For instance, let’s say that your friend is out walking her dog. She measures her distances travelled as 7 m, then 2 m. What is her total distance travelled?

Our initial measurements (7 m and 2 m) are not very precise - that is, the person measuring the 7 m may have measured 6.9 m (off by 10 cm) and rounded up, or 7.3 m (off by 30 cm) and rounded down… we just don’t know. All we do know is that the measurement was somewhere around 7 m.

Had the number been 7.000 m, however, all those zeroes imply a much more precise measurement. In this case, if there was any rounding, it might have been down from 7.004 m (0.4 cm), or up from 7.000 9 m (0.1 cm) - much smaller amounts than above!

So it doesn’t make sense to say that our total distance is 7 m + 2 m = 9.000 m. We cannot assume the final answer (9.000 m) is more precise than either of the initial measurements.

This is a common mistake made when dividing. If we divide our two distances, to determine a ratio, many people would write \frac{2m}{7m}= 0.285714285

But because of the imprecision in the 2 m and 7 m, our answer to the same precision should be only 0.3 (regardless of what the calculator tells us). Never just write down all the numbers the calculator gives you.

So how do we know just how much precise our answer should be?

The final answer can only be as precise as the least precise measurement used in the calculation. If we had calculated 7 m + 2.000 m, there is an improvement in the precision, but the final answer would still be 9 m, because we are only as good as our least precise quantity.

We determine precision by counting significant figures, and ensuring that our answer has the same number of significant figures as the measurement with the least number of significant figures. Watch this video to get a sense of the rules when it comes to significant figures:

Re-watch the video as many times as you like. Create a reflection, and record the answers to the following questions:

  1. What difference does a decimal point make? What is the difference in precision between a number like 940 and 940.0?
  2. Does it make sense that a number like 83 400 is more precise than 7.3? Why or why not?

Think about it!

Consider a track and field event such as a 400 m race.

  1. What is each runner’s total distance? Is it the same for each athlete? How do you know?
  2. What is each runner’s total displacement from the start of the race to the finish line? Is it the same for each athlete?

Each runner is covering a distance of 400 m - it would have to be the same for each athlete, or it wouldn’t be a fair race, would it? You may have noticed that the runners each start at a separate line. Because the runners travel in an oval, if they all started at the same line, the runners on the outside of the oval would end up travelling farther than those running on the inside of the oval. The staggered start ensures each runner runs exactly 400 m. If they ran the race in a straight line, they would all have started from the same place.

Each runner’s displacement from start to finish, though, is more interesting! For the athlete in the inner lane (lane 1), who starts at the finish line, her displacement is 0 m - she ends where she began. The runner in the outer lane (lane 8), however, ends 53 m behind where she started. The runners in lanes 2 through 7 will all have a displacement behind where they started, the magnitude of which will depend on their starting position. In no lane, though, is the runner’s displacement 400 m; no runner ends 400 m from their starting position.