an ant walks 22 cm in a northerly directions turns and walks southeasterly (45 degree East Of South) a distance of 47 cm what is a resultant displacement?
33.23cm
see below
To find the resultant displacement, we can break down the ant's movements into its northward and southeastward components.
Step 1: Northward displacement:
The ant walks 22 cm in a northerly direction. This is the magnitude of the northward displacement.
Step 2: Southeastward displacement:
The ant walks 47 cm in a direction that is 45 degrees east of south. A 45-degree angle east of south corresponds to a 45-degree angle below the negative y-axis.
To find the southeastward displacement, we need to find the eastward and southward components of this movement.
The eastward component (in the x-direction) can be found using trigonometry:
eastward component = magnitude of displacement * cos(angle)
eastward component = 47 cm * cos(45°)
The southward component (in the y-direction) can also be found using trigonometry:
southward component = magnitude of displacement * sin(angle)
southward component = 47 cm * sin(45°)
Step 3: Calculating the resultant displacement:
The resultant displacement is the vector sum of the northward and southeastward components.
We can represent these components as vectors and add them to find the resultant displacement:
Resultant Displacement = Northward Displacement + Southeastward Displacement
Magnitude of Resultant Displacement = sqrt((Northward Displacement)^2 + (Southeastward Displacement)^2)
Let's calculate the values:
Magnitude of Northward Displacement = 22 cm
Magnitude of Southeastward Displacement:
eastward component = 47 cm * cos(45°)
southward component = 47 cm * sin(45°)
Resultant Displacement = sqrt((22 cm)^2 + (eastward component)^2 + (southward component)^2)
To find the resultant displacement of the ant, we need to calculate the vector sum of its northward and southeasterly displacements.
Let's break down the magnitudes and directions of the two displacements:
1. Northward displacement: The ant walks 22 cm in a northerly direction. Since this is a straight line, the magnitude of this displacement is simply 22 cm, and the direction is straight north.
2. Southeasterly displacement: The ant walks 47 cm in a southeasterly direction, which is 45 degrees east of south. To calculate the components of this displacement, we need to use trigonometry. The angle between the displacement and the southern direction (0 degrees) is 45 degrees, so we can use the cosine and sine functions:
Magnitude of the south component = 47 cm * cos(45 degrees)
Magnitude of the east component = 47 cm * sin(45 degrees)
Now, let's calculate the magnitudes of the components:
Magnitude of the south component = 47 cm * cos(45 degrees) = 47 cm * (sqrt(2)/2) = 33.24 cm
Magnitude of the east component = 47 cm * sin(45 degrees) = 47 cm * (sqrt(2)/2) = 33.24 cm
To determine the direction of the resultant displacement, we can use trigonometry as well. The angle between the resultant displacement and the northern direction (0 degrees) can be found using the tangent function:
tan(angle) = (magnitude of the east component) / (magnitude of the south component)
angle = arctan((magnitude of the east component) / (magnitude of the south component))
angle = arctan(33.24 cm / 33.24 cm) = arctan(1) = 45 degrees
Finally, to calculate the magnitude of the resultant displacement, we can use the Pythagorean theorem:
Magnitude of the resultant displacement = sqrt((magnitude of the north component)^2 + (magnitude of the south component)^2)
Magnitude of the resultant displacement = sqrt((22 cm)^2 + (33.24 cm)^2) = sqrt(484 cm^2 + 1102.74 cm^2) = sqrt(1586.74 cm^2) = 39.85 cm
Therefore, the resultant displacement of the ant is approximately 39.85 cm at an angle of 45 degrees north of east.