A man walks five miles due east and then three miles in a direction 60° north of east .calculate the displacement of the man.
I made a sketch and got triangle ABC, with angle B = 120°, AB = 5, and BC = 3
looks like a normal cosine law problem
AC^2 = 5^2 + 3^2 - 2(5)(3)cos120°
....
then use the sine law to find angle A
To calculate the displacement of the man, we can use the concept of vector addition.
Step 1: Convert the distances to vectors.
A vector has both magnitude (distance traveled) and direction. Let's represent the 5-mile walk due east as vector A and the 3-mile walk in the direction 60° north of east as vector B.
Vector A: 5 miles due east (→)
Vector B: 3 miles at a direction 60° north of east (↖)
Step 2: Break the vectors into their components.
The east direction is our reference direction (x-axis), so we need to find the components of each vector along the x-axis and the y-axis.
For vector A:
The x-component of vector A is 5 miles (since it is entirely in the east direction).
The y-component of vector A is 0 miles (since it is entirely in the east direction).
For vector B:
To find the x-component of vector B, we use the cosine of the angle between the vector and the x-axis:
x-component = magnitude of B * cos(60°) = 3 miles * cos(60°) = 1.5 miles
To find the y-component of vector B, we use the sine of the angle between the vector and the x-axis:
y-component = magnitude of B * sin(60°) = 3 miles * sin(60°) = 2.6 miles (rounded to one decimal place)
So, the components of vector B are:
x-component = 1.5 miles
y-component = 2.6 miles
Step 3: Add the components of the vectors.
To find the displacement, we add the corresponding components of both vectors.
Displacement in the x-direction = x-component of A + x-component of B = 5 miles + 1.5 miles = 6.5 miles
Displacement in the y-direction = y-component of A + y-component of B = 0 miles + 2.6 miles = 2.6 miles
Step 4: Calculate the magnitude and direction of displacement.
The magnitude of the displacement (D) is given by the formula:
D = sqrt(Displacement in the x-direction^2 + Displacement in the y-direction^2)
So, D = sqrt((6.5 miles)^2 + (2.6 miles)^2) = sqrt(42.25 + 6.76) = sqrt(49.01) = 7 miles (rounded to one decimal place)
The direction of the displacement can be found using the tangent of the angle:
tan(angle) = y-component of displacement / x-component of displacement
So, tan(angle) = 2.6 miles / 6.5 miles
Therefore, angle = arctan(2.6/6.5) = arctan(0.4)
Thus, the direction of the displacement is approximately 22.6 degrees north of east.
Thus, the displacement of the man is 7 miles at an angle of approximately 22.6 degrees north of east.