Answer:
displacement = ( 4.045 km , 72 degrees )
Step-by-step explanation:
Given:
- The radius if the circular path r = 2.5 km
- Couple hikes 70 % of the circular track.
Find:
What is the displacement for the couple's hiking trip? Give the direction in terms of an angle measured CCW from the Easterly direction.
Solution:
- The start point of the journey is west and due south in a circular direction.
- First we will determine the 70 % of the circumference of the circular path traveled. Using the arc length S formula:
S = 0.70* ( 2*pi*r) = 7/5 pi*r
- The arc length calculated above is the distance traveled. However, we are interested in the "displacement" which is the shortest distance from start point to 70% of the track.
- The total amount of angle swept between the two points in radians is:
S = r*Q
Q = S / r = 7/5*pi
- The remaining angle angle to complete the full circle:
Q' = 2*p - Q = 2*p -7/5*pi
Q' = 0.6*pi.
- The displacement between two points can be calculated using sine rule as follows:
sin(Q') / displacement = sin (0.2*pi) / r
displacement = r*sin(Q') / sin (0.2*pi)
displacement = 2.5*sin(0.6*pi) / sin (0.2*pi)
displacement = 4.045 km
- The position with respect to the East direction is:
pi - Q' = pi - 0.6*pi = 0.4*pi
Convert into degrees: 0.4*pi * 180 /pi = 72 degrees