A particle take the following successive displacement 10km south west, 50 KM north of East. Find the magnitude of the resistant
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem.
Let's call the displacement in the South-West direction as A and the displacement in the North-East direction as B.
The magnitude of the resultant displacement (R) can be found using the formula:
R^2 = A^2 + B^2
The displacement of 10km in the South-West direction can be split into two components: 10km towards the south and 10km towards the west. The displacement of 50km in the North-East direction can be split into two components: 50km towards the north and 50km towards the east.
So, A = 10km + 50km (towards west)
= 10km + (-50km)
= -40km (towards west)
B = 50km + 10km (towards north)
= 60km (towards north)
Now, substituting the values in the formula, we get:
R^2 = (-40km)^2 + (60km)^2
R^2 = 1600km^2 + 3600km^2
R^2 = 5200km^2
Taking the square root of both sides, we get:
R = sqrt(5200km^2)
R ≈ 72.11km
Therefore, the magnitude of the resultant displacement is approximately 72.11km.
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem.
First, let's represent the successive displacements as vectors:
- The displacement of 10 km south-west can be represented as a vector S with a magnitude of 10 km and an angle of 45 degrees south-west.
- The displacement of 50 km north of East can be represented as a vector E with a magnitude of 50 km and an angle of 90 degrees east.
To find the resultant displacement, we need to add the vectors S and E.
Step 1: Convert the angles to their equivalent Cartesian coordinates:
The angle 45 degrees south-west is equivalent to an angle of 45 degrees clockwise from the positive x-axis. Therefore, the coordinates are (cos(45°), -sin(45°)).
The angle 90 degrees east is equivalent to an angle of 90 degrees counter-clockwise from the positive x-axis. Therefore, the coordinates are (cos(90°), sin(90°)).
Step 2: Calculate the Cartesian coordinates of the resultant vector:
The x-coordinate of the resultant vector is obtained by adding the x-coordinates of S and E:
x-coordinate = cos(45°) + cos(90°) = √2/2 + 0 = √2/2
The y-coordinate of the resultant vector is obtained by adding the y-coordinates of S and E:
y-coordinate = -sin(45°) + sin(90°) = -√2/2 + 1 = 1 - √2/2
Step 3: Calculate the magnitude of the resultant vector:
The magnitude of the resultant vector is given by the equation:
magnitude = √(x-coordinate² + y-coordinate²)
Substituting the values:
magnitude = √( (√2/2)² + (1 - √2/2)² )
= √(2/4 + 1 - √2/2 + 1/4)
= √(9/4 - √2/2)
Therefore, the magnitude of the resultant displacement is √(9/4 - √2/2).