An ice skater skates 16.9 m southwest, then
33.8 m east, and finally 50.7 m in a direction
50.9◦ north of east.
Find the magnitude of the displacement re-
quired to bring the skater back to her starting
point.
Answer in units of m
To find the magnitude of the displacement required to bring the skater back to her starting point, we need to consider the vectors representing her movements.
The first movement is 16.9 m southwest, which can be represented as a vector (-16.9 m, -16.9 m) since southwest is in the negative x and y direction.
The second movement is 33.8 m east, which can be represented as a vector (33.8 m, 0 m) since it is in the positive x direction.
The third movement is 50.7 m in a direction 50.9° north of east. We can break this into its x and y components. The x component can be calculated as (50.7 m * cos(50.9°)), and the y component can be calculated as (50.7 m * sin(50.9°)).
Adding up all the x and y components of the movements, we get:
X component = 33.8 m + (50.7 m * cos(50.9°))
Y component = -16.9 m + (50.7 m * sin(50.9°))
To find the magnitude of the displacement, we can use the Pythagorean theorem:
Magnitude = sqrt((X component)^2 + (Y component)^2)
Plugging in the values and calculating, we get:
Magnitude = sqrt((33.8 m + (50.7 m * cos(50.9°)))^2 + (-16.9 m + (50.7 m * sin(50.9°)))^2)
Calculating this expression will give us the magnitude of the displacement required to bring the skater back to her starting point.
To find the magnitude of the displacement required to bring the ice skater back to her starting point, we need to calculate the resultant displacement.
Step 1: Convert the given distances and angles into vector components.
The skater first skates 16.9 m southwest, which can be broken down into two components:
- The westward component is -16.9 m.
- The southward component is -16.9 m.
Next, the skater skates 33.8 m east, which has only an eastward component:
- The eastward component is 33.8 m.
Finally, the skater skates 50.7 m in a direction 50.9° north of east. To find the components, we need to split this displacement into eastward (x-axis) and northward (y-axis) components:
- The eastward component is 50.7 m * cos(50.9°).
- The northward component is 50.7 m * sin(50.9°).
Step 2: Calculate the sum of all the components.
Adding up the components, we have:
- The total eastward component is 33.8 m + 50.7 m * cos(50.9°).
- The total northward component is -16.9 m + 50.7 m * sin(50.9°).
Step 3: Find the magnitude of the resultant displacement.
The magnitude of the resultant displacement is given by the Pythagorean theorem:
Resultant displacement = sqrt[(total eastward component)^2 + (total northward component)^2]
Substituting the calculated components into the equation, we get:
Resultant displacement = sqrt[(33.8 m + 50.7 m * cos(50.9°))^2 + (-16.9 m + 50.7 m * sin(50.9°))^2]
Calculating this expression will give us the magnitude of the displacement required to bring the skater back to her starting point.
16.9m[225o]
33.8m[0o]
50.7m[50.9o]
X=16.9*cos225+33.8+50.7*cos50.9=53.8 m.
Y=16.9*sin225+0+50.7*sin50.9 = 27.4 m.
tan A = Y/X = 27.4/53.8 = 0.50921
A = 27o
D = X/cos A = 53.8/cos27 = 60.38 m.