A jogger follows this route every morning. Assume each segment is a straight line.
A= 1000m due south
B= 500m at 53 degrees east of south
C= 400m due east
What is the jogger's displacement from start to end including magnitude and direction?
Express as vectors
A =[0, -1000]
B =[500sin(53°), -500cos(53°)]
C =[400, 0]
Sum the vectors, then calculate the magnitude and direction of that sum.
To find the jogger's displacement, we need to calculate the vector sum of each segment. The magnitude of the displacement can be found using the Pythagorean theorem, and the direction can be determined using trigonometry.
Let's break down the problem step by step:
1. Segment A: 1000m due south
The vector for segment A can be represented as (0, -1000) since it is going only in the vertical direction.
2. Segment B: 500m at 53 degrees east of south
To represent segment B, we need to find the vertical and horizontal components of this vector. We can find the vertical component by multiplying the magnitude (500m) by the sine of the angle (53 degrees): 500 * sin(53) ≈ 402.17m. The horizontal component can be found by multiplying the magnitude by the cosine of the angle: 500 * cos(53) ≈ 321.57m.
Thus, the vector for segment B can be represented as (321.57, -402.17).
3. Segment C: 400m due east
As segment C is entirely in the horizontal direction, its vector can be represented as (400, 0), since it is only moving to the right.
Now, let's find the vector sum by adding the corresponding components:
Vertical component: -1000 + (-402.17) + 0 = -1402.17m
Horizontal component: 0 + 321.57 + 400 = 721.57m
The displacement vector can now be represented as (721.57, -1402.17), where the horizontal component represents the magnitude in the east-west direction and the vertical component represents the magnitude in the north-south direction.
To find the magnitude of the displacement, we can use the Pythagorean theorem:
Magnitude = sqrt((721.57)^2 + (-1402.17)^2) ≈ 1577.86m
Finally, to find the direction, we can use trigonometry:
Direction = arctan(-1402.17 / 721.57) ≈ -61.68 degrees
Therefore, the jogger's displacement from start to end has a magnitude of approximately 1577.86m and a direction of approximately -61.68 degrees (measured clockwise from the positive x-axis).