A very strong girl runs 200 miles east for several hours. After a brief rest she runs another 100 miles north. She stops for the night at a local bed and breakfast and the next morning walks 75 miles directly south-east. Find the magnitude and direction of her displacement.
I can do that in weeks or months, not hours!
Total displacement is the sum of the components of individual displacements. Most of the time, displacements are resolved in x (or east) and y (north) directions.
200 miles -> east = (200,0)
100 miles -> north = (0,100)
75 miles -> S/E = (75√(1/2),-75√(1/2))
Add up the three vectors to get:
(253,147), i.e.
253 miles towards east, 147 towards north.
Alternatively, it can be combined to form a single vector (use Pythagoras theorem):
257.4 miles in the direction NθE where θ=arctan(253/147)=59.8° approx.
To find the displacement of the girl, we need to determine the straight-line distance and direction between her starting point and ending point.
Let's break down the given information step by step:
1. The girl runs 200 miles east. This means she moves horizontally along the east direction.
2. After a rest, the girl runs another 100 miles north. This means she moves vertically along the north direction.
3. The girl stops at a bed and breakfast for the night, which doesn't contribute to her displacement.
4. The next morning, the girl walks 75 miles directly southeast. This results in a diagonal movement combining both southeast and downhill.
To determine the overall displacement, we can use vector addition. Let's assign directions to the movements:
- East will be denoted as (+x) direction.
- North will be denoted as (+y) direction.
- South will be denoted as (-y) direction.
- Southeast will be a diagonal movement combining (+x) and (-y) directions.
Based on this, we can break down the displacements:
- The 200 miles run east has a displacement of (+200, 0).
- The 100 miles run north has a displacement of (0, +100).
- The 75 miles walked southeast has a displacement of (+75(cos45°), -75(sin45°)).
Using trigonometric identities, we can calculate the values for cos45° and sin45°:
- cos45° = sin45° = (√2)/2
Calculating the displacement:
Displacement = (200 + 75(√2)/2, 100 - 75(√2)/2)
= (200 + 75(√2)/2, 100 - 75(√2)/2)
Now, let's calculate the magnitude of the displacement using the Pythagorean theorem:
Magnitude = √[(200 + 75(√2)/2)^2 + (100 - 75(√2)/2)^2]
= √[40000 + 15000(√2) + 16875/2 + 5625/2]
= √[40000 + 7500(√2) + 11250 + 3750]
= √[65000 + 8250(√2)]
= √[65000 + 11678.48]
= √76678.48
≈ 277.10 miles (rounded to two decimal places)
To find the direction of displacement, we need to determine the angle it makes with the x-axis. Using trigonometry:
Direction = arctan[(100 - 75(√2)/2) / (200 + 75(√2)/2)]
= arctan(0.293)
≈ 16.28° (rounded to two decimal places)
Therefore, the magnitude of the girl's displacement is approximately 277.10 miles, and the direction is approximately 16.28° east of north.