A person walks 21 m East and then walks 34 m at an angle 32◦ North of East. What is the magnitude of the total displacement?
Answer in units of m.
Disp. = 21 + 34m[32o].
Disp. = (21+34*Cos32) + (34*sin32)i = 49.8 + 18.0i.
Disp. = sqrt(49.8^2+18^2)
Well, I don't want to sound too dis-placed, but let's calculate it!
To find the total displacement, we can consider the East direction as the positive x-axis and the North direction as the positive y-axis.
The person walked 21 m East, so their displacement in the x-direction would be 21 m.
Next, they walked 34 m at an angle of 32 degrees North of East. We can use a little bit of trigonometry to find their displacement in the y-direction:
Displacement in the y-direction = 34 m * sin(32 degrees)
So, the person's displacement in the y-direction would be approximately 17.32 m (rounded to two decimal places).
Now, let's calculate the total displacement by using the Pythagorean theorem:
Total Displacement = √((displacement in x-direction)^2 + (displacement in y-direction)^2)
Total Displacement = √((21 m)^2 + (17.32 m)^2)
Total Displacement ≈ √(441 m² + 300 m²)
Total Displacement ≈ √741 m²
Total Displacement ≈ 27.2 m (rounded to one decimal place)
So, the magnitude of the total displacement is approximately 27.2 m. Keep on walking, my friend!
To find the magnitude of the total displacement, we need to combine the two displacements given in both magnitude and direction.
First, let's break down the given displacements into their x and y components.
Displacement 1: 21 m East
- The x-component is 21 m (since it is in the East direction), and the y-component is 0 m (since it is along the x-axis).
Displacement 2: 34 m at an angle 32° North of East
- To find the x and y components, we'll use trigonometry.
- The x-component is given by: 34 m * cos(32°)
- The y-component is given by: 34 m * sin(32°)
Now, let's calculate the x and y components for Displacement 2:
- x-component = 34 m * cos(32°) ≈ 28.858 m
- y-component = 34 m * sin(32°) ≈ 18.209 m
Next, let's combine the x and y components from both displacements.
- The x-component of the total displacement is the sum of the x-components from each displacement: 21 m + 28.858 m ≈ 49.858 m
- The y-component of the total displacement is the sum of the y-components from each displacement: 0 m + 18.209 m ≈ 18.209 m
Now, we can find the magnitude of the total displacement using the Pythagorean theorem:
magnitude of total displacement = √(x-component² + y-component²)
Substituting the values:
magnitude of total displacement = √(49.858 m² + 18.209 m²) ≈ √(2486.136 m² + 332.126 m²)
magnitude of total displacement ≈ √(2818.262 m²) ≈ 53.088 m
Therefore, the magnitude of the total displacement is approximately 53.088 meters.