A person walks the path shown below. The total trip consists of four straight-line paths.
Path 1 is 40.0 m due east.
Path 2 is 170.0 m due south.
Path 3 is 120.0 m 30.0° south of west.
Path 4 is 190.0 m 60.0° north of west.
At the end of the walk, what is the person's resultant displacement?
( meters and degrees south of west)
Well, let's take a clownish approach to solving this problem!
First, let's consider the paths as a series of circus acts.
Path 1 can represent a clown juggling 40 flaming torches, successfully heading due east without burning themselves.
Path 2 is like a daring tightrope walk 170 meters above the ground, but instead of going forward, they manage to go due south without falling off. Impressive!
Path 3 is a trapeze act swinging 120 meters away south of west with a 30-degree angle. Talk about acrobatics!
Path 4 is a high-flying cannonball act, blasting 190 meters in the air at a 60-degree angle north of west. That's some serious clown catapulting!
Now, let's calculate the resultant displacement, or where our wandering clown ended up.
To find the resultant displacement, we need to add all the individual displacements together. Let's crunch some numbers!
First, we need to break down Path 3 and 4 into their east-west and north-south components.
For Path 3:
The east-west component is 120.0 m * cos(30°) = 103.92 m westward.
The north-south component is 120.0 m * sin(30°) = 60.00 m southward.
And for Path 4:
The east-west component is 190.0 m * cos(60°) = 95.00 m westward.
The north-south component is 190.0 m * sin(60°) = 164.98 m northward.
Now, let's add them all up:
Total east-west displacement = 40.0 m (Path 1) - 103.92 m (Path 3) - 95.00 m (Path 4) = -159.92 m westward
Total north-south displacement = -170.0 m (Path 2) + 60.00 m (Path 3) + 164.98 m (Path 4) = 54.98 m northward
So, the person's resultant displacement is approximately 159.92 meters westward and 54.98 meters northward.
If we want to express it as meters and degrees south of west, we need to find the angle.
To find the angle, we can use the tangent function:
Angle (θ) = tan^(-1) (north-south displacement / east-west displacement)
Plugging in the numbers:
θ = tan^(-1) (54.98 m / 159.92 m)
θ ≈ 18.08° south of west
Therefore, the person's resultant displacement is approximately 159.92 meters westward and 54.98 meters northward, at an angle of about 18.08° south of west.
That's quite a comedic journey our clown has embarked on!
To find the person's resultant displacement, we need to find the x-component and the y-component of the displacement and then calculate the magnitude and direction of the displacement.
1. Path 1 is 40.0 m due east.
The x-component of Path 1 is 40.0 m.
2. Path 2 is 170.0 m due south.
The y-component of Path 2 is -170.0 m (negative because it's south).
3. Path 3 is 120.0 m 30.0° south of west.
To find the x-component of Path 3, we use cosine:
x-component of Path 3 = 120.0 m * cos(30°) = 120.0 m * 0.866 = 103.92 m (rounded to two decimal places)
To find the y-component of Path 3, we use sine:
y-component of Path 3 = 120.0 m * sin(30°) = 120.0 m * 0.5 = 60.0 m
4. Path 4 is 190.0 m 60.0° north of west.
To find the x-component of Path 4, we use cosine:
x-component of Path 4 = 190.0 m * cos(60°) = 190.0 m * 0.5 = 95.0 m
To find the y-component of Path 4, we use sine:
y-component of Path 4 = 190.0 m * sin(60°) = 190.0 m * 0.866 = 164.14 m (rounded to two decimal places)
Now, we can sum up the x-components and y-components to find the resultant displacement:
x-component = 40.0 m + 103.92 m + 95.0 m = 238.92 m (rounded to two decimal places)
y-component = -170.0 m + 60.0 m + 164.14 m = 54.14 m (rounded to two decimal places)
To find the magnitude of the displacement, we use the Pythagorean theorem:
magnitude = √(x-component^2 + y-component^2)
magnitude = √(238.92^2 + 54.14^2) = √(57125.91 + 2934.43) = √60060.34 = 245.17 m (rounded to two decimal places)
To find the direction of the displacement, we use inverse tangent (tan⁻¹):
direction = tan⁻¹(y-component / x-component)
direction = tan⁻¹(54.14 m / 238.92 m) = 12.93° (rounded to two decimal places)
Therefore, the person's resultant displacement is 245.17 m at 12.93° south of west.
To find the resultant displacement of the person, we need to calculate the vector sum of all four paths.
Path 1: The person walks 40.0 m due east. This can be represented as a vector, V1 = 40.0 m due east.
Path 2: The person walks 170.0 m due south. This can be represented as a vector, V2 = 170.0 m due south.
Path 3: The person walks 120.0 m 30.0° south of west. To represent this vector, we need to break it down into its components. The horizontal component (x-axis) can be calculated as 120.0 m * cos(30.0°) and the vertical component (y-axis) can be calculated as 120.0 m * sin(30.0°). So, V3 = (120.0 m * cos(30.0°))i - (120.0 m * sin(30.0°))j.
Path 4: The person walks 190.0 m 60.0° north of west. Again, we need to break this vector down into its components. The horizontal component (x-axis) can be calculated as 190.0 m * cos(60.0°) and the vertical component (y-axis) can be calculated as -190.0 m * sin(60.0°) since it is in the opposite direction. So, V4 = (190.0 m * cos(60.0°))i - (190.0 m * sin(60.0°))j.
Now, let's add all these vectors together to find the resultant displacement (R).
R = V1 + V2 + V3 + V4
Now, we can calculate the horizontal and vertical components separately:
Horizontal component (x-axis):
Rx = (V1x + V2x + V3x + V4x)
= 40.0 m + 0 + (120.0 m * cos(30.0°)) + (190.0 m * cos(60.0°))
Vertical component (y-axis):
Ry = (V1y + V2y + V3y + V4y)
= 0 + (-170.0 m) + (-120.0 m * sin(30.0°)) + (-190.0 m * sin(60.0°))
To find the magnitude (R) and direction (θ) of the resultant displacement, we can use Pythagoras' theorem and inverse tangent function:
R = sqrt(Rx^2 + Ry^2)
θ = atan(Ry/Rx)
Now, plug in the values and calculate the resultant displacement.
40E = (40,0)
170S = (0,-170)
120W30°S = (-103.9,-60)
190W60°N = (-95,165.5)
Add them up to get (31.1,-64.5) = 71.6m W64°S