# Question 8f515

Nov 15, 2016

Distance walked = $160 m$ Displacement = $70.7 m$

#### Explanation:

You need to know the difference between distance and displacement.

Distance is probably easier to start with. It simply means "How far did you walk altogether?"

There were 3 different stages: 50m, then 80m, then 30m

$50 + 80 + 30 = 160 m$ is the total distance walked.

Displacement means "How far away from the starting point was I when I stopped walking?

You should realise that walking east cancels out some of the distance to the west.

$< - - - - - - - - -$
$\text{ } 80 m$ west

$\text{ } - - - > < - - - - -$
$\text{ } 30 m$ east $\text{ } 50 m$ west

That means that your final position is the same as walking

50 m North and then 50 m West.

These lines form 2 sides of a right-angled isosceles triangle.

You need to know the length of the longest side - the slanted line which goes directly from start to finish. It is called the hypotenuse of the triangle.

To find its length, you can draw a scale diagram and measure the length of the longest side.

However, there is a method called "Pythagoras's Theorem" which you can use to calculate that length..

${\text{length}}^{2} = {50}^{2} + {50}^{2}$

${\text{length}}^{2} = 5000$

$\text{length} = \sqrt{5000}$

$\text{length} = 70.7 m$

This length is the displacement, because you end up 70.7 m away from the starting place.

Nov 15, 2016

Just a note about how to deal with $\sqrt{5000}$ without a calculator.

#### Explanation:

$\sqrt{5000} \to \sqrt{50 \times 100} \to \sqrt{50 \times {10}^{2}}$

But $\sqrt{{10}^{2}} = 10$ so take ${10}^{2}$ outside the root giving:

$10 \sqrt{50}$

............................................................................................................

Ok lets see if there are any squared numbers in 50 that we can also take outside the root.

$50 = 5 \times 10$

But $10 = 2 \times 5$ so we have

$50 = 5 \times 5 \times 2 = {5}^{2} \times 2$
...................................................................................................................

$10 \sqrt{50} = 10 \sqrt{{5}^{2} \times 2} = 10 \times 5 \sqrt{2} = 50 \sqrt{2}$

We can not take this any further as 2 is a prime number

So $\sqrt{5000} = 50 \sqrt{2} \text{ }$ as an exact value.

As an approximate value this is 70.71 to 2 decimal places.