A golfer takes three putts to get the ball into the hole. The first putt displaces the ball 3.32 m north, the second 1.63 m southeast, and the third 0.703 m southwest. What are (a) the magnitude and (b) the angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east?

Question

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball 3.32 m north, the second 1.63 m southeast, and the third 0.703 m southwest. What are (a) the magnitude and (b) the angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east?

Answer 1

To solve this problem, we can break down the displacements into their respective components and add them up:

First putt:
- Displacement: 3.32 m north
- Components: Δx = 0 m (east-west direction), Δy = 3.32 m (north-south direction)

Second putt:
- Displacement: 1.63 m southeast
- Components: Δx = 1.63 cos(45°) m (east-west direction), Δy = -1.63 sin(45°) m (north-south direction)

Third putt:
- Displacement: 0.703 m southwest
- Components: Δx = -0.703 cos(45°) m (east-west direction), Δy = -0.703 sin(45°) m (north-south direction)

Now, we can add up the components:
Δx_total = Δx_first + Δx_second + Δx_third
Δy_total = Δy_first + Δy_second + Δy_third

Δx_total = 0 m + 1.63 cos(45°) m + (-0.703 cos(45°)) m
Δy_total = 3.32 m + (-1.63 sin(45°)) m + (-0.703 sin(45°)) m

Calculating the values, we find:
Δx_total ≈ 0.884 m
Δy_total ≈ 0.456 m

(a) The magnitude of the displacement needed to get the ball into the hole in just one putt can be calculated using the Pythagorean theorem:
magnitude = sqrt(Δx_total^2 + Δy_total^2)

magnitude ≈ sqrt((0.884 m)^2 + (0.456 m)^2)
magnitude ≈ sqrt(0.7805 m^2 + 0.2079 m^2)
magnitude ≈ sqrt(0.9884 m^2)
magnitude ≈ 0.994 m

Therefore, the magnitude of the displacement needed to get the ball into the hole in just one putt is approximately 0.994 m.

(b) The angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east can be calculated using trigonometry:
angle = arctan(Δy_total / Δx_total)

angle = arctan(0.456 m / 0.884 m)
angle ≈ arctan(0.516)
angle ≈ 27.6°

Therefore, the angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east is approximately 27.6°.