a very strong girl runs 200 miles 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 ealks 75 miles directly south-east. Find the magnitude and direction of her displacement? (answer)
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To find the magnitude and direction of the girl's displacement, we need to combine the distances and directions of her individual movements.
First, let's calculate the total distance she ran: 200 miles + 100 miles = 300 miles.
Next, let's visualize her movements on a Cartesian coordinate system. Consider the north direction as positive and the south direction as negative along the y-axis, and the east direction as positive and the west direction as negative along the x-axis.
For the first leg of her journey, the strong girl ran 200 miles for several hours. Since no direction is specified, let's assume she ran along the x-axis, which means her displacement is purely horizontal.
For the second leg, she ran another 100 miles north. This means her displacement is purely vertical along the positive y-axis.
For the third leg, she walked 75 miles directly southeast. This means her displacement is at a 45-degree angle southeast from the positive x-axis.
Now, to find her overall displacement, we will use vector addition. We need to find the resultant vector by adding all the individual displacement vectors.
First, we need to consider the horizontal component of her displacement. Since her first movement was along the x-axis, her horizontal displacement is 200 miles.
Next, the vertical component of her displacement is given by the second leg of her journey, which is 100 miles in the positive y-axis direction.
To find the diagonal displacement from the third leg, we can calculate the horizontal and vertical components. Using a right-angled triangle, we can determine that the horizontal component is 75 miles multiplied by cos(45°) ≈ 52.92 miles, and the vertical component is 75 miles multiplied by sin(45°) ≈ 52.92 miles.
Now, we can calculate the total horizontal and vertical components by adding up the respective components of all three legs:
Horizontal component: 200 miles + 0 miles + 52.92 miles = 252.92 miles (east)
Vertical component: 0 miles + 100 miles + 52.92 miles = 152.92 miles (north)
Finally, we can use the Pythagorean theorem to find the magnitude of her displacement. The magnitude is given by the square root of the sum of the squares of the horizontal and vertical components:
Magnitude = sqrt((252.92)^2 + (152.92)^2) ≈ 287.7 miles
To find the direction of her displacement, we can use trigonometry. The angle can be determined using the inverse tangent function:
Angle = arctan(152.92/252.92) ≈ 30.12 degrees
Therefore, the girl's displacement is approximately 287.7 miles at an angle of 30.12 degrees northeast from the positive x-axis.