A golfer, putting on a green, requires three strokes to "hole the ball." During the first putt, the ball rolls 5.9 m due east. For the second putt, the ball travels 2.4 m at an angle 20° north of east. The third putt is 0.50 m due north. What displacement (magnitude and direction relative to due east) would have been needed to "hole the ball" on the very first putt?

Question

A golfer, putting on a green, requires three strokes to "hole the ball." During the first putt, the ball rolls 5.9 m due east. For the second putt, the ball travels 2.4 m at an angle 20° north of east. The third putt is 0.50 m due north. What displacement (magnitude and direction relative to due east) would have been needed to "hole the ball" on the very first putt?

Answer 1

East

5.9 + 2.4 cos 20 = 8.16
north
2.4 sin 20 + .5 = 1.32

d^2 = 8.16^2 + 1.32^2

tan heading = 8.16/1.32
heading east of north = 80.8 deg
or
heading north of east = 90-80.8 = 9.19 deg

Answer 2

To find the displacement needed on the first putt, we need to analyze the total displacement of the golf ball.

Displacement is a vector quantity and has both magnitude and direction. In this case, direction is relative to due east.

Let's break down the given information step by step and calculate the total displacement:

1. First putt: The ball rolls 5.9 m due east.
- This means the first putt has a displacement of +5.9 m due east.

2. Second putt: The ball travels 2.4 m at an angle 20° north of east.
- To calculate the horizontal displacement, we need to find the component of the second putt in the eastward direction.
- The eastward component of the second putt can be found by multiplying the total distance (2.4 m) by the cosine of the angle (20°).
- Eastward component = 2.4 m * cos(20°) = 2.4 m * 0.9397 ≈ 2.255 m due east.

3. Third putt: The ball travels 0.50 m due north.
- The third putt does not contribute to the displacement in the eastward direction.

Now, we add up the eastward displacements from each putt:

Total eastward displacement = 5.9 m + 2.255 m = 8.155 m due east.

So, to hole the ball on the very first putt, a displacement of approximately 8.155 m due east would have been needed.

Answer 3

To find the displacement required to "hole the ball" on the very first putt, we can add up the individual displacements of each putt.

First, let's break down the second putt into its eastward and northward components. The eastward component can be found by calculating the cosine of the given angle:

Eastward component = magnitude of second putt * cosine(angle)
= 2.4 m * cos(20°)

Northward component = magnitude of second putt * sine(angle)
= 2.4 m * sin(20°)

Now we can calculate the total eastward displacement by adding up the distances traveled in each putt:

Total eastward displacement = first putt distance + second putt eastward component + third putt distance
= 5.9 m + 2.4 m * cos(20°) + 0 m (as the third putt is due north)

To find the magnitude and direction of the displacement, we can use the Pythagorean theorem and trigonometry.

Magnitude of displacement = square root of (Total eastward displacement)^2 + (northward component)^2

Direction of displacement = arctan(northward component / Total eastward displacement) + 90° (to add the angle Ture north)

Let's calculate the values:

Total eastward displacement = 5.9 m + 2.4 m * cos(20°) + 0 m
= 5.9 m + 2.4 m * 0.9397 + 0 m
= 5.9 m + 2.2553 m
≈ 8.16 m

Northward component = 2.4 m * sin(20°)
= 2.4 m * 0.3420
≈ 0.82 m

Magnitude of displacement = square root of (8.16 m)^2 + (0.82 m)^2
≈ square root of (66.7056 m^2 + 0.6724 m^2)
≈ square root of 67.378 m^2
≈ 8.21 m

Direction of displacement = arctan(0.82 m / 8.16 m) + 90°
≈ arctan(0.1005) + 90°
≈ 5.77° + 90°
≈ 95.77°

Therefore, the magnitude of the displacement required to "hole the ball" on the very first putt is approximately 8.21 meters, and the direction relative to due east is approximately 95.77°.