One afternoon, a couple walks five-sixths of the way around a circular lake, the radius of which is 1.45 km. They start at the west side of the lake and head due south to begin with.
(a) What is the distance they travel?
(b) What are the magnitude and direction (relative to due east) of the couple's displacement?
distance=5/6 * 2PI *r
displacement: final point-initial point
Displacement= r@5/6 *360 - r@0
so in circular coordinates, displacement is r@300-r@0
Now in rectangular coordinates
displcement=r (sin60 N + cos60W)-r(sin0 N + cos0 W)
and you can work that out.
To find the distance the couple travels, we need to calculate the length of the arc they walk.
(a) The arc length, given the angle in radians, can be calculated using the formula:
Arc length = radius * angle
In this case, the couple walks five-sixths of the way around the lake, which is equivalent to an angle of 5/6 * 2π radians (the full circle is 2π radians).
Arc length = 1.45 km * (5/6 * 2π) radians
To compute the distance, we multiply the arc length by the radius of the lake:
Distance = 1.45 km * (5/6 * 2π) radians
Simplifying the equation:
Distance = 1.45 km * (5/3) * π ≈ 7.63 km
Therefore, the distance the couple travels is approximately 7.63 km.
(b) To find the magnitude and direction of the couple's displacement, we can use vector addition. Since they start at the west side and head due south, their displacement is the vector sum of their eastward and northward components.
The eastward component is given by:
Eastward displacement = radius * cos(angle)
Since they walk five-sixths of the way around the lake, the angle is 5/6*2π radians. Thus:
Eastward displacement = 1.45 km * cos(5/6*2π)
The northward component can be found using:
Northward displacement = radius * sin(angle)
Therefore:
Northward displacement = 1.45 km * sin(5/6*2π)
To calculate the magnitude and direction, we use the Pythagorean theorem:
Magnitude of displacement = sqrt((Eastward displacement)^2 + (Northward displacement)^2)
Direction = arctan(Northward displacement / Eastward displacement)
Substituting the values, we can compute:
Magnitude of displacement = sqrt((1.45 km * cos(5/6*2π))^2 + (1.45 km * sin(5/6*2π))^2)
Direction = arctan((1.45 km * sin(5/6*2π)) / (1.45 km * cos(5/6*2π)))
Evaluating these equations will give you the magnitude and direction of the couple's displacement.