A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 3.90 m to the north, d2 = 2.06 m northeast, and d3 = 1.14 m at 30.0° west of south (see the figure below). Starting at the same initial point, an expert golfer could make the hole in what single displacement?

magnitude: m
direction: ° north of east

To find the single displacement that an expert golfer would need to make the hole, we can determine the total displacement made by the novice golfer and then subtract it from the final position.

First, we need to break down the displacements into their respective components. The direction and magnitude of the displacements are given as follows:

d1: 3.90 m north
d2: 2.06 m northeast
d3: 1.14 m at 30.0° west of south

We'll convert the angles to their respective components using trigonometry. The northeast direction is 45 degrees from both north and east, so it will have equal north and east components. The angle 30.0° west of south will have a south component and an east component.

Now, let's calculate the components of each displacement:

d1:
North component: 3.90 m
East component: 0 m

d2:
North component: sin(45°) * 2.06 m (approximately)
East component: cos(45°) * 2.06 m (approximately)

d3:
South component: sin(30°) * 1.14 m (approximately)
East component: cos(30°) * 1.14 m (approximately)

Next, we can find the total components of the displacements by adding up the respective north and east components:

Total North component = d1 North component + d2 North component + d3 South component
Total East component = d2 East component + d3 East component

Now, let's calculate the total north and east components:

Total North component = 3.90 m + sin(45°) * 2.06 m - sin(30°) * 1.14 m (approximately)
Total East component = cos(45°) * 2.06 m + cos(30°) * 1.14 m (approximately)

Once we have the total north and east components, we can calculate the magnitude and direction of the single displacement.

Magnitude = sqrt((Total North component)^2 + (Total East component)^2)
Direction = arctan(Total East component / Total North component)

Finally, we can plug in the values and solve for the magnitude and direction of the single displacement.

To find the single displacement made by the expert golfer, we can add the individual displacements together using vector addition.

First, let's convert the given displacement d2 from northeast to its north and east components.

The north component (d2_north) can be found using cosine:
d2_north = d2 * cos(45°) = 2.06 m * cos(45°) = 1.46 m

The east component (d2_east) can be found using sine:
d2_east = d2 * sin(45°) = 2.06 m * sin(45°) = 1.46 m

Now let's calculate the total north and east components by adding the respective components from the three displacements:

Total north component (d_north) = d1 + d2_north + d3 * sin(30°) = 3.90 m + 1.46 m + 1.14 m * sin(30°)
Total east component (d_east) = d2_east + d3 * cos(30°) = 1.46 m + 1.14 m * cos(30°)

To find the magnitude and direction of the total displacement, we can use the Pythagorean theorem and inverse trigonometric functions:

Magnitude (d) = sqrt(d_north^2 + d_east^2)

Direction (θ) = atan2(d_east, d_north)

Let's calculate the values:

Magnitude (d) = sqrt((3.90 m + 1.46 m + 1.14 m * sin(30°))^2 + (1.46 m + 1.14 m * cos(30°))^2)

Direction (θ) = atan2(1.46 m + 1.14 m * cos(30°), 3.90 m + 1.46 m + 1.14 m * sin(30°))

Calculating the values:

Magnitude (d) = sqrt((3.90 m + 1.46 m + 1.14 m * 0.5)^2 + (1.46 m + 1.14 m * 0.87)^2)

Magnitude (d) = sqrt((3.90 m + 1.46 m + 0.57 m)^2 + (1.46 m + 0.99 m)^2)

Magnitude (d) = sqrt((6.93 m)^2 + (2.45 m)^2)

Magnitude (d) = sqrt(48.03 m^2 + 6.01 m^2)

Magnitude (d) = sqrt(54.04 m^2)

Magnitude (d) = 7.35 m

Direction (θ) = atan2(1.46 m + 1.14 m * 0.87, 3.90 m + 1.46 m + 1.14 m * 0.5)

Direction (θ) = atan2(1.46 m + 0.99 m, 6.93 m + 1.46 m + 0.57 m)

Direction (θ) = atan2(2.45 m, 9.96 m)

Direction (θ) = atan(2.45 m / 9.96 m)

Direction (θ) ≈ 14.6°

Therefore, an expert golfer could make the hole in a single displacement of approximately 7.35 m magnitude and 14.6° north of east.

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