A boy runs 10.3 blocks North, 6.1 blocks
Northeast, and 5 blocks West.
Determine the length of the displacement
vector that goes from the starting point to his
final position.
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 ◦
.
D = 10.3[90o] + 6.1[45] + 5[180].
X = 10.3*Cos90 + 6.1*Cos45 + 5*Cos180 =
0 + 4.31 - 5 = -0.69 Blocks.
Y = 10.3*sin90 + 6.1*sin45 + 5*sin180 =
10.3 + 4.31 + 0 = 14.61 Blocks.
D = sqrt(X^2+Y^2) = sqrt((-0.69)^2 + 14.61^2) = 14.6 Blocks.
Tan A = Y/X = 14.61/-0.69 = -87.3o =
87.3o N. of W. = 92.7o CCW.
To determine the length of the displacement vector, we need to find the magnitude of the vector formed by the net displacement. The length of the displacement vector can be found using the Pythagorean theorem, given by:
|d| = √(dx² + dy²)
where dx and dy are the displacements along the horizontal and vertical axes, respectively.
In this case, the boy runs 5 blocks to the west, which gives a horizontal displacement of dx = -5. He also runs 10.3 blocks to the north and 6.1 blocks northeast. To find the vertical displacement dy, we need to decompose the northeast displacement into its vertical and horizontal components.
Let's represent the northeast displacement as a right-angled triangle, where the hypotenuse represents the 6.1 blocks northeast displacement, the vertical leg represents the north displacement, and the horizontal leg represents the east displacement. Since the boy is moving in the first quadrant (northeast), both the vertical and horizontal components are positive.
Using the right triangle, we can apply the Pythagorean theorem to find the vertical and horizontal components:
(6.1 blocks)² = (dx)² + (dy)²
Solving for dy:
(dy)² = (6.1 blocks)² - (dx)²
(dy)² = (6.1 blocks)² - (-5 blocks)²
(dy)² = 37.21 blocks² - 25 blocks²
(dy)² = 12.21 blocks²
dy = sqrt(12.21 blocks²)
dy ≈ 3.495 blocks
The vertical component (dy) represents the north displacement. Therefore, the net vertical displacement (10.3 blocks north + 3.495 blocks north) is given by: dely = 10.3 blocks + 3.495 blocks ≈ 13.795 blocks.
Now we can calculate the magnitude of displacement (|d|):
|d| = sqrt((dx)² + (dy)²)
|d| = sqrt((-5 blocks)² + (13.795 blocks)²)
|d| = sqrt(25 blocks² + 190.722025 blocks²)
|d| = sqrt(215.722025 blocks²)
|d| ≈ 14.68 blocks
Therefore, the length of the displacement vector from the starting point to the final position is approximately 14.68 blocks.
To determine the direction of the displacement vector, we can find the angle it makes with the east direction. We can use trigonometry to calculate this angle. Let's call this angle θ.
tan(θ) = (dy) / (dx)
tan(θ) = (13.795 blocks) / (-5 blocks)
θ = arctan((13.795 blocks) / (-5 blocks))
θ ≈ -69.12°
Since we are using counterclockwise as the positive angular direction, and the angle is measured from the east, the direction of the displacement vector is approximately -69.12°.
Therefore, the length of the displacement vector is approximately 14.68 blocks, and the direction is approximately -69.12°.
To determine the length of the displacement vector, we can use the Pythagorean theorem. The displacement vector is the straight-line distance between the starting point and the final position.
First, let's break down the boy's movements into their components. He runs 10.3 blocks North, 6.1 blocks Northeast, and 5 blocks West.
The North component is simply 10.3 blocks.
To find the Northeast component, we need to split it into its North and East components using trigonometry. The angle between Northeast and North is 45 degrees (since Northeast is halfway between North and East). Using the cosine function, we can find the North component of the Northeast movement:
North component of Northeast = 6.1 blocks * cos(45°) ≈ 4.315 blocks.
The West component is simply 5 blocks.
Next, we can find the total North component by adding the North component of the initial movement (10.3 blocks) and the North component of the Northeast movement (4.315 blocks):
Total North component = 10.3 blocks + 4.315 blocks = 14.615 blocks.
The total East component is found by subtracting the West component (5 blocks) from the East component of the Northeast movement. Since the angle between Northeast and East is 45 degrees, the East component of the Northeast movement is the same as the North component. Therefore:
Total East component = 4.315 blocks - 5 blocks = -0.685 blocks.
Now, we can apply the Pythagorean theorem to find the length of the displacement vector:
Displacement vector length = √(Total North component^2 + Total East component^2)
= √((14.615 blocks)^2 + (-0.685 blocks)^2)
≈ √(213.498225 blocks^2 + 0.470225 blocks^2)
≈ √213.968 blocks^2
≈ 14.628 blocks (rounded to the nearest thousandth).
Therefore, the length of the displacement vector is approximately 14.628 blocks.
Now, let's determine the direction of the displacement vector. We need to find the angle between the displacement vector and the East direction, measured counterclockwise.
We can use the arctan function to find this angle:
Angle = arctan(Total North component / Total East component)
= arctan(14.615 blocks / -0.685 blocks)
≈ arctan(-21.342)
≈ -86.983°.
Since we are measuring counterclockwise from East, we take the negative of the angle to keep it within the range of -180° to +180°.
Therefore, the direction of the displacement vector is approximately -86.983°.