Determine the magnitude and direction of the displacement if a man walks 32.5 km 45° north of east, and then walks due east 19.5 km.
R=48,2
To determine the magnitude and direction of the displacement, we can break down the man's motion into horizontal and vertical components.
First, let's calculate the horizontal and vertical displacements separately.
Horizontal Displacement:
The man walks 32.5 km in a direction 45° north of east.
To find the horizontal displacement, we need to find the component of this distance in the eastward direction.
Using trigonometry, we can find the eastward component by multiplying the distance by the cosine of the angle:
Horizontal Displacement = 32.5 km × cos(45°)
Horizontal Displacement = 32.5 km × 0.7071
Horizontal Displacement = 23.03675 km (rounded to the nearest thousandth)
Vertical Displacement:
The man walks 32.5 km in a direction 45° north of east.
To find the vertical displacement, we need to find the component of this distance in the northward direction.
Using trigonometry, we can find the northward component by multiplying the distance by the sine of the angle:
Vertical Displacement = 32.5 km × sin(45°)
Vertical Displacement = 32.5 km × 0.7071
Vertical Displacement = 23.03675 km (rounded to the nearest thousandth)
Now, let's find the total displacement by adding the horizontal and vertical displacements.
Total Displacement:
Total Displacement = √(horizontal displacement)^2 + (vertical displacement)^2
Total Displacement = √(23.03675 km)^2 + (23.03675 km)^2
Total Displacement = √(529.293755625 km^2 + 529.293755625 km^2)
Total Displacement = √(1058.58751125 km^2)
Total Displacement = 32.568 km (rounded to the nearest thousandth)
Direction of Displacement:
To find the direction of the displacement, we can use trigonometry.
The angle of the displacement, with respect to the eastward direction, can be found using the tangent:
Direction of Displacement = arctan(vertical displacement / horizontal displacement)
Direction of Displacement = arctan(23.03675 km / 23.03675 km)
Direction of Displacement = arctan(1)
Direction of Displacement ≈ 45° (rounded to the nearest degree)
Therefore, the magnitude of the displacement is approximately 32.568 km, and the direction of the displacement is approximately 45° east of north.
To determine the magnitude and direction of the displacement, we can break down the man's movements into two components: the northward component and the eastward component.
Given that the man walks 32.5 km 45° north of east, we can use trigonometry to find the northward and eastward components of this movement.
The northward component can be found using the sine function:
Northward component = 32.5 km * sin(45°) = 32.5 km * 0.7071 ≈ 23.01 km
The eastward component can be found using the cosine function:
Eastward component = 32.5 km * cos(45°) = 32.5 km * 0.7071 ≈ 23.01 km
So, the northward and eastward components of the first movement are both approximately equal to 23.01 km.
Next, the man walks due east for 19.5 km. Since this movement is directly east, there is no northward component.
Now, to find the total displacement, we can add the northward and eastward components together.
Magnitude of displacement = √((23.01 km)^2 + (19.5 km)^2) ≈ √(529.82 km^2 + 380.25 km^2) ≈ √910.07 km^2 ≈ 30.17 km
The magnitude of the displacement is approximately 30.17 km.
To find the direction of the displacement, we can use trigonometry and the inverse tangent function.
Direction = atan(Northward component / Eastward component)
Direction = atan(23.01 km / 19.5 km) ≈ atan(1.179808) ≈ 50.69°
The direction of the displacement is approximately 50.69°, measured counterclockwise from the east direction.
Therefore, the magnitude of the displacement is approximately 30.17 km, and the direction is approximately 50.69° counterclockwise from the east.
Add the vectors. Break them up into components.
32.5km @45deg= 32.5*sin45 N + 32.5*cos 45 E
add that to 19.5E
32.5*sin45N + (32.5*.707+19.5)E
I assume you can take if from here.