A novice golfer on the green takes three strokes to sink the ball. The successive displacements are 3.80 m to the north, 1.90 m northeast, and 1.10 m at 30.0° west of south. Starting at the same initial point, an expert golfer could make the hole in what single displacement?
(need angle in degrees north of east)
How do you find the angle of resultant vector?
Can you show the work please bob
To find the single displacement that an expert golfer would take to make the hole, we can combine the three successive displacements of the novice golfer.
Let's break down the information given:
1. The novice golfer takes three strokes, each with a certain displacement.
a) The first displacement is 3.80 m to the north.
b) The second displacement is 1.90 m northeast.
c) The third displacement is 1.10 m at 30.0° west of south.
2. We need to find a single displacement for the expert golfer.
To combine the displacements, we can use vector addition. We will add up the individual displacements using the properties of vectors (magnitude and direction).
First, let's convert the second and third displacements into their north and east components.
For the second displacement (1.90 m northeast):
The north component = 1.90 m * cos(45°) (since northeast is 45° from both north and east)
= 1.90 m * cos(45°)
= 1.90 m * 0.7071
= 1.3457 m north
The east component = 1.90 m * sin(45°)
= 1.90 m * sin(45°)
= 1.90 m * 0.7071
= 1.3457 m east
So, the second displacement can be represented as 1.3457 m north + 1.3457 m east.
For the third displacement (1.10 m at 30.0° west of south):
The south component = 1.10 m * cos(30°) (since 30° west of south is 30° from south)
= 1.10 m * cos(30°)
= 1.10 m * 0.866
= 0.953 m south
The west component = 1.10 m * sin(30°)
= 1.10 m * sin(30°)
= 1.10 m * 0.5
= 0.55 m west
So, the third displacement can be represented as 0.953 m south + 0.55 m west.
Now, let's add up the individual displacements:
Total north component = 3.80 m north + 1.3457 m north + 0.953 m south (combine north and south components)
= 3.80 m + 1.3457 m - 0.953 m
= 4.1927 m north
Total east component = 1.3457 m east - 0.55 m west (combine east and west components)
= 1.3457 m - 0.55 m
= 0.7957 m east
Now, we have the total displacement for the expert golfer: 4.1927 m north + 0.7957 m east.
To find the angle north of east, we can use trigonometry.
angle = tan^(-1)(north component / east component)
= tan^(-1)(4.1927 m / 0.7957 m)
≈ 80.9°
Therefore, an expert golfer could make the hole with a single displacement of approximately 4.19 m at an angle of 80.9° north of east.