person walks 50 m east, turns south and walks 32 m, then turns to the southeast (45° south of east) and walks 44 m. a)Determine the magnitude of the resultant displacement vector in meters. b) Determine the direction of the total displacement. Give your response in terms of a compass direction and the angle from it in degrees. c) If the person does this in 52 s, what was the magnitude of their average velocity in m/s?
I figured the other parts out, but Im just confused about the magnitude, would I just add everything up
If we label the successive locations as A,B,C,D, then if we set A = (0,0) we have
B = A + (50,0)
C = B + (0,-32)
D = C + (44/√2,-44/√2) = (81.11,-62.11)
(a) 102.16
(b) E37.4°S
(c) 102.6m/52s = 1,97 m/s
To determine the magnitude of the resultant displacement vector and the direction of the total displacement, we can break down each movement into its respective components.
Step 1: Walking 50 m east.
- East component: 50 m
- North component: 0 m
Step 2: Turning south and walking 32 m.
- East component: 0 m
- South component: -32 m
Step 3: Turning southeast (45° south of east) and walking 44 m.
To find the east and south components, we need to calculate the components of the displacement vector using trigonometry.
- East component: 44 m * cos(45°) = 31.1127 m
- South component: 44 m * sin(45°) = 31.1127 m
Now, we can find the total displacement by adding up the components:
- East component: 50 m + 31.1127 m = 81.1127 m
- North component: 0 m + (-32 m) + 31.1127 m = -0.8873 m (since south is negative)
a) The magnitude of the resultant displacement vector is:
Resultant displacement = sqrt((81.1127)^2 + (-0.8873)^2) ≈ 81.12 m
b) The direction of the total displacement can be found using trigonometry:
Angle = arctan((81.1127 / -0.8873))
Angle ≈ -89.430°
Since the angle is negative, we need to convert it to a compass direction. In this case, it would be:
Total displacement = 89.430° south of west
c) To determine the magnitude of the average velocity, we need to calculate the total distance traveled and the total time taken.
Total distance traveled = 50 m + 32 m + 44 m = 126 m
Total time taken = 52 s
Average velocity = Total distance traveled / Total time taken
Average velocity = 126 m / 52 s ≈ 2.42 m/s
To determine the magnitude of the resultant displacement vector, we need to find the total distance traveled in the east and south directions:
1. Start by breaking down the initial 50 m east movement into its x and y components.
- The x component is 50 m (as it was in the east direction), and the y component is 0 m.
2. Next, consider the movement south for 32 m. This will only affect the y component.
- Subtract 32 m from the y component, resulting in a y component of -32 m (negative because it's in the south direction).
3. Now, let's consider the movement in the southeast direction. We'll need to break this down into its x and y components.
- The direction is 45° south of east, which means the angle with the positive x-axis is (90° - 45°) = 45°.
- The distance traveled is 44 m.
- Use trigonometry to find the x and y components:
- The x component = 44 m * cos(45°) ≈ 31.1 m
- The y component = 44 m * sin(45°) ≈ 31.1 m
4. To find the total x and y components, add up the values from the previous steps.
- x component: 50 m + 31.1 m = 81.1 m (east)
- y component: -32 m + 31.1 m = -0.9 m (south)
5. Finally, calculate the magnitude of the resultant displacement vector using the Pythagorean theorem:
- Magnitude = √(x component^2 + y component^2)
- Magnitude = √(81.1^2 + (-0.9)^2) ≈ 81.1 m
Therefore, the magnitude of the resultant displacement vector is approximately 81.1 meters (m).
Now let's determine the direction of the total displacement:
6. Calculate the angle of the resultant displacement vector (θ) using inverse tangent (tan^-1).
- θ = tan^-1 (y component / x component)
- θ = tan^-1(-0.9 m / 81.1 m)
- θ ≈ -0.64°
7. Convert the angle to compass direction:
- The angle is approximately -0.64°, which means it lies between west and northwest.
- Since it's a negative angle, we'll subtract it from 360° to find the direction from the north.
- Direction from the north = 360° - 0.64° ≈ 359.36°
Therefore, the direction of the total displacement is approximately 359.36°, which corresponds to almost due north.
Lastly, let's determine the magnitude of the average velocity:
8. Average velocity is defined as the total displacement divided by the total time taken.
- Magnitude of average velocity = Magnitude of resultant displacement vector / time
- Magnitude of average velocity = 81.1 m / 52 s ≈ 1.56 m/s
Therefore, the magnitude of the person's average velocity is approximately 1.56 meters per second (m/s).