A boy runs 18.4 blocks North, 12.9 blocks
Northeast, and 19.7 blocks West.
a)Determine the length of the displacement
vector that goes from the starting point to his final position.
b)Determine the direction of the displacement vector. Use counterclockwise as the positive
angular direction, between the limits of
−180◦ and +180◦ measured from East.
Answer in units of degrees
To determine the length of the displacement vector, we can use the Pythagorean theorem as the boy's displacement involves movements in two perpendicular directions (North and West).
a) Using the Pythagorean theorem:
Displacement^2 = (North)^2 + (West)^2
Displacement^2 = (18.4)^2 + (-19.7)^2 (negative sign for West direction)
Displacement^2 = 338.56 + 388.09
Displacement^2 = 726.65
Taking the square root of both sides gives us:
Displacement = √(726.65)
Displacement ≈ 26.97 blocks
Therefore, the length of the displacement vector is approximately 26.97 blocks.
b) To determine the direction of the displacement vector, we can use inverse tangent (arctan) function.
Direction = arctan(North/West)
Direction = arctan(18.4/(-19.7))
Direction ≈ -41.87 degrees
Since the direction is measured counterclockwise as the positive angular direction and between the limits of −180◦ and +180◦ measured from East, the direction is approximately -41.87 degrees.
Therefore, the direction of the displacement vector is approximately -41.87 degrees.
To determine the length of the displacement vector, we can use the Pythagorean theorem. The displacement vector is the straight-line distance from the starting point to the final position.
a) To find the length of the displacement vector, we can calculate the total distance traveled in the North and Northeast directions, subtracted by the distance traveled in the West direction.
The distance traveled North is 18.4 blocks.
The distance traveled Northeast is 12.9 blocks.
To calculate the North and Northeast distance, we can use the Pythagorean theorem:
North and Northeast distance = √(18.4^2 + 12.9^2)
North and Northeast distance ≈ 22.11 blocks
The distance traveled West is 19.7 blocks.
So, the length of the displacement vector is:
Displacement length = North and Northeast distance - West distance
Displacement length = 22.11 blocks - 19.7 blocks
Displacement length ≈ 2.41 blocks
b) To determine the direction of the displacement vector, we can use trigonometry. We can calculate the angle between the displacement vector and the East direction.
Using the tangent function, we can calculate the angle:
Tangent of the angle = (North and Northeast distance) / (West distance)
Angle = arctan((North and Northeast distance) / (West distance))
Angle = arctan(22.11 / 19.7)
Angle ≈ 48.6 degrees
Since the angle is measured counterclockwise from the East direction, the direction of the displacement vector is approximately 48.6 degrees counterclockwise from the East.