Question

id: period date distance vs displacement pythagorean theorem (a^2 + b^2 = c^2) define distance: displacement: instructions: a. draw a diagram for the following situations. b. calculate both distance and displacement. 1. john runs 10 miles north, then turns around and runs 20 miles south. distance: displacement: 2. ana runs 5 laps around a 1000 meter track. distance: displacement: 3. john runs 3 meters east, and then turns north and runs 4 meters. distance: displacement: 4. ryan runs 30 meters north, then turns west and runs 20 meters, and finally turns and runs 30 meters south. distance: displacement: page 1

Explanation:

Problem 1: John runs 10 miles north, then turns around and runs 20 miles south.

Step 1: Calculate Distance

Distance is the total path length. So we add the two distances.
$10 + 20 = 30$ miles.

Step 2: Calculate Displacement

Displacement is the straight - line distance from the start to the end point. Running 10 miles north and then 20 miles south means the net displacement is $20 - 10=10$ miles south (or we can say the magnitude is 10 miles in the south direction).

Step 1: Calculate Distance

Distance is the total length of the path. Each lap is 1000 meters, and she runs 5 laps. So we multiply the length of one lap by the number of laps.
$5\times1000 = 5000$ meters.

Step 2: Calculate Displacement

Displacement is the straight - line distance from the start to the end point. Since she ends up at the same point where she started (after 5 laps around a track), the displacement is 0 meters.

Step 1: Calculate Distance

Distance is the total path length. We add the two distances.
$3 + 4=7$ meters.

Step 2: Calculate Displacement

We can use the Pythagorean theorem to find the displacement. The eastward and northward motions form the two legs of a right - triangle, and the displacement is the hypotenuse. Let $a = 3$ meters (east), $b = 4$ meters (north). Then by the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5$ meters. The direction is northeast (or we can calculate the angle $\theta=\tan^{- 1}(\frac{4}{3})$ from the east direction towards the north, but for magnitude, it's 5 meters).

Answer:

• Distance: 30 miles
• Displacement: 10 miles south (or 10 miles in the south direction)

Problem 2: Ana runs 5 laps around a 1000 - meter track.