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 displacement vector, we need to consider both the distance and direction traveled by the boy.
a) The displacement vector represents the straight-line distance between the starting point and the final position. To find it, we can use the Pythagorean theorem:
The distance traveled North = 18.4 blocks
The distance traveled Northeast = 12.9 blocks
The distance traveled West = 19.7 blocks
The displacement vector is the net sum of these distances. We can calculate it using the formula:
displacement = √(north^2 + northeast^2 + west^2)
Plugging in the given values:
displacement = √(18.4^2 + 12.9^2 + (-19.7)^2)
displacement ≈ √(338.56 + 166.41 + 388.09)
displacement ≈ √893.06
displacement ≈ 29.86 blocks
So, the length of the displacement vector is approximately 29.86 blocks.
b) To determine the direction of the displacement vector, we can use trigonometry. We need to find the angle the displacement vector makes with the East direction.
We can calculate the angle using the inverse tangent (arctan) function:
angle = arctan(north/east)
Plugging in the given values:
angle = arctan((18.4+12.9)/(-19.7))
angle ≈ arctan(31.3/-19.7)
angle ≈ -57.58 degrees
Since the answer is given in the counterclockwise direction, we have a negative angle because it's a clockwise direction when measuring from East.
Therefore, the direction of the displacement vector is approximately -57.58 degrees.