as she pick up her riders a bus driver traverses four successive displacements represented by the expression (-6.30 b)i-(4.00 b cos 40)i-(4.00 sin 40)j+(3.00 b cos 50)i-(3.00 b sin 50)j-(5.00 b)j here b represents one city block,a convenient unit of distance of uniform size; i is east; and j is north. the displacements at 40 degree and 50 degree represent travel on roadways in the city that are at these angles to the main east-west and north-south streets.(a)draw a map of the successive displacements(b)what total distance did she travel? (c)compute the magnitude and direction of her total displacements.
PLEASE ANSWERS MY ASSIGNMENT
If I read your somewhat confusing vectors correctly, they are (in units of b)
-6.30i
-4.00cos40i - 4.00sin40j
3.00cos50i - 3.00sin50j
-5.00j
the distances traveled in each case is
6.30
4.00
3.00
5.00
adding up to 18.30
the directions are all given. just read 'em off
what the answer to letter c. how can i find the answer please illustrate to me
adding up the i and j values, the total displacement is
-12.4i - 4.87j = 13.32 @ 158.6°
|-12.4i - 4.87j| = √(12.4^2 + 4.87^2)
tan158.6° = -4.87/-12.4
how did u get -12.4i and 4.87j
To solve this problem, we will break it down into three parts: drawing a map, calculating the total distance traveled, and finding the magnitude and direction of the total displacement.
a) Drawing the map of the successive displacements:
Let's represent each displacement as a vector on a coordinate system. The east direction is represented by the x-axis (i) and the north direction by the y-axis (j). We can draw the vectors by aligning their tails and connecting the heads. Here's the step-by-step process:
1. Start at the origin (0,0).
2. Displacement (-6.30 b)i represents moving 6.30 blocks in the west direction. So, draw a vector pointing left (west) for 6.30 units.
3. Displacement (-4.00 b cos 40)i represents moving 4.00 blocks in the southwest direction. Using the cosine function, calculate the x-component of this displacement: 4.00 * cos(40) = -3.06 b. Draw a vector that extends 3.06 units to the left (west).
4. Displacement (-4.00 sin 40)j represents moving 4.00 blocks in the south direction. Using the sine function, calculate the y-component of this displacement: 4.00 * sin(40) = -2.57 b. Draw a vector that extends 2.57 units downwards (south).
5. Displacement (3.00 b cos 50)i represents moving 3.00 blocks in the northeast direction. Using the cosine function, calculate the x-component of this displacement: 3.00 * cos(50) = 1.93 b. Draw a vector that extends 1.93 units to the right (east).
6. Displacement (-3.00 b sin 50)j represents moving 3.00 blocks in the south direction. Using the sine function, calculate the y-component of this displacement: 3.00 * sin(50) = -2.29 b. Draw a vector that extends 2.29 units downwards (south).
7. Displacement (-5.00 b)j represents moving 5.00 blocks in the south direction. Draw a vector that extends 5.00 units downwards (south).
b) Calculating the total distance traveled:
To find the total distance traveled, we need to find the magnitude (length) of each vector. Then, we will sum up these magnitudes. The distance between the start and end point is not relevant here since we are only interested in the total distance traveled.
1. Magnitude of (-6.30 b)i = 6.30 b
2. Magnitude of (-3.06 b)i = 3.06 b
3. Magnitude of (-2.57 b)j = 2.57 b
4. Magnitude of (1.93 b)i = 1.93 b
5. Magnitude of (-2.29 b)j = 2.29 b
6. Magnitude of (-5.00 b)j = 5.00 b
Summing up the magnitudes: 6.30 b + 3.06 b + 2.57 b + 1.93 b + 2.29 b + 5.00 b = 21.15 b
So, the total distance traveled is 21.15 city blocks.
c) Finding the magnitude and direction of the total displacement:
To find the magnitude of the total displacement, we need to find the resultant vector by summing up all the individual vectors:
Resultant vector = (-6.30 b)i + (-3.06 b)i + (-2.57 b)j + (1.93 b)i + (-2.29 b)j + (-5.00 b)j
Combining like terms:
Resultant vector = (-6.3 - 3.06 + 1.93)b i + (-2.57 - 2.29 - 5.00)b j
Resultant vector = (-7.43 b) i + (-9.86 b) j
The magnitude of the resultant vector can be found using the Pythagorean theorem:
Magnitude = sqrt((-7.43 b)^2 + (-9.86 b)^2)
= sqrt(55.1449 b^2 + 97.2196 b^2)
= sqrt(152.3645 b^2)
= 12.345 b
The magnitude of the resultant vector is 12.345 city blocks.
To find the direction of the resultant vector, we can use trigonometry. Its angle (θ) with the positive x-axis can be calculated as:
θ = atan((-9.86 b) / (-7.43 b))
= atan(9.86 / 7.43)
≈ 50.53 degrees
Since the negative signs cancel out, the direction is opposite to the orientation of the vectors on the y-axis. Thus, the direction of the total displacement is approximately 50.53 degrees south of the west.
In summary:
a) Draw a map of the successive displacements as described above.
b) The total distance traveled is 21.15 city blocks.
c) The magnitude of the total displacement is 12.345 city blocks, and its direction is approximately 50.53 degrees south of the west.