One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 1.03 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 is the magnitude of the couple’s displacement? (c) What is the direction (relative to due east) of the couple’s displacement?
Distance=0.75• 2•π•R=0.75•2•π•1.03=4.85 km.
Displacement is =2√R=2√1.03=2.03 km
direction 45ºrestectively - x-axis direction
or 135º respectively +x-direction
Displacement is 1.03^2*1.03^2=c^2 the third side is your displacement
To calculate the distance the couple traveled, we need to find the circumference of the lake.
(a) The circumference of a circle is found using the formula:
Circumference = 2 * π * Radius
Given that the radius of the lake is 1.03 km, the distance they traveled is:
Distance = (3/4) * Circumference
= (3/4) * (2 * π * 1.03 km)
= 2.44 km
Therefore, the distance they traveled is 2.44 km.
(b) To find the magnitude of the couple's displacement, we need to calculate the straight-line distance from their starting point to the ending point.
The straight-line distance is the diameter of the circle since they have traveled three-fourths of the way around.
Magnitude of Displacement = Diameter = 2 * Radius
= 2 * 1.03 km
= 2.06 km
Therefore, the magnitude of the couple's displacement is 2.06 km.
(c) The direction of the couple's displacement relative to due east can be found by drawing a line connecting the starting point (west side) and ending point.
Since they were initially heading due south (opposite to due north), the direction of the couple's displacement is to the east of due south.
Therefore, the direction of the couple's displacement (relative to due east) is south-east.
To find the answers to these questions, we can use some concepts of geometry and trigonometry:
(a) To determine the distance they travel, we need to calculate the length of three-fourths of the circumference of the circular lake. The formula for the circumference of a circle is given by: C = 2πr, where C is the circumference and r is the radius.
In this case, the radius is 1.03 km. So, the circumference is:
C = 2π(1.03) km ≈ 6.46 km
Three-fourths of the circumference is then:
Distance traveled = (3/4) * 6.46 km ≈ 4.85 km
Therefore, the couple travels approximately 4.85 km.
(b) The magnitude of the couple's displacement refers to the straight-line distance between the starting and ending points. In this case, they walk from the west side to the south side of the lake, covering an angle of 90 degrees.
To find the displacement, we can use the formula for the length of an arc on a circle: s = rθ, where s is the arc length, r is the radius, and θ is the central angle.
In this case, the angle is 90 degrees (or π/2 radians), and the radius is 1.03 km. Therefore, the displacement is:
Displacement = 1.03 km * π/2 ≈ 1.62 km
Therefore, the magnitude of the couple's displacement is approximately 1.62 km.
(c) To determine the direction of the couple's displacement relative to due east, we need to consider the angle formed by the displacement with the eastward direction. Since they start on the west side and head due south, the angle would be in the southeast direction.
To find this angle, we can use trigonometry. In a right triangle with the adjacent side as the horizontal displacement and the hypotenuse as the displacement, we can use the tangent function: tan(θ) = Opposite/Adjacent.
In this case, the opposite side is the horizontal displacement (eastward) and the adjacent side is the displacement itself. Therefore:
tan(θ) = eastward displacement/1.62 km
Solving for θ, we get:
θ ≈ arctan(eastward displacement/1.62 km)
The approximate value depends on the exact eastward displacement, which would be the radius minus the horizontal displacement.
Calculating this value will give us the direction of the couple's displacement relative to due east.