A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 3.98 m to the north, d2 = 2.01 m northeast, and d3 = 1.12 m at 30.0° west of south (see the figure). Starting at the same initial point, an expert golfer could make the hole in what single displacement?
d1 = (0,3.98)
d2 = (2.01/√2,2.01/√2)
d3 = (-1.12/2,-1.12√3/2)
just add them up and convert back to compass directions.
To find the single displacement of the expert golfer, we can add the given displacements together:
d1 = 3.98 m to the north
d2 = 2.01 m northeast
d3 = 1.12 m at 30.0° west of south
First, let's determine the vertical and horizontal components of each displacement:
For d1:
Vertical component: 3.98 m x sin(90°) = 3.98 m
Horizontal component: 3.98 m x cos(90°) = 0
For d2:
Vertical component: 2.01 m x sin(45°) = 1.42 m
Horizontal component: 2.01 m x cos(45°) = 1.42 m
For d3:
Vertical component: 1.12 m x sin(30.0°) = 1.12 m x (-0.5) = -0.56 m
Horizontal component: 1.12 m x cos(30.0°) = 1.12 m x (0.866) = 0.97 m
Now, let's add the vertical and horizontal components separately:
Vertical component: 3.98 m + 1.42 m + (-0.56 m) = 4.84 m
Horizontal component: 0 m + 1.42 m + 0.97 m = 2.39 m
Finally, we can find the magnitude of the single displacement using the Pythagorean theorem:
Magnitude = sqrt(Vertical component^2 + Horizontal component^2)
Magnitude = sqrt((4.84 m)^2 + (2.39 m)^2) ≈ 5.43 m
Therefore, the expert golfer could make the hole in a single displacement of approximately 5.43 meters.
To find the single displacement to make the hole in one stroke, we need to analyze the given displacements and combine them into a single displacement.
First, let's analyze the given displacements one by one:
1. The first displacement, d1 = 3.98 m to the north, is straightforward. It is a pure north direction, so it only has vertical displacement.
2. The second displacement, d2 = 2.01 m northeast, has both horizontal and vertical components. We can find the horizontal and vertical displacements separately using trigonometry.
Let's assume the angle between the northeast direction and the positive x-axis is theta.
The horizontal component (dx) can be found using the cosine function: dx = d2 * cos(theta).
The vertical component (dy) can be found using the sine function: dy = d2 * sin(theta).
Considering that theta = 45 degrees for northeast direction, we have dx = dy = d2 / √2.
3. The third displacement, d3 = 1.12 m at 30.0° west of south, is at an angle measured from the south. To find the horizontal and vertical components, we need to decompose this displacement.
Let's assume the angle with the negative y-axis is phi.
The horizontal component (dx) can be found using the sine function as dx = d3 * sin(phi).
The vertical component (dy) can be found using the cosine function as dy = d3 * cos(phi).
Considering that phi = 150 degrees for a 30 degrees angle west of south, we have dx = -d3 * sin(150) and dy = -d3 * cos(150).
Now, let's combine all the displacements to find the resultant displacement:
To combine the displacements, we need to add the horizontal and vertical components separately. In vector form, the resultant displacement r can be written as:
r = (Σ dx, Σ dy).
Adding the horizontal components:
Σ dx = 0 (from d1) + d2 / √2 (from d2) + (-d3 * sin(150)) (from d3).
Adding the vertical components:
Σ dy = d1 (from d1) + d2 / √2 (from d2) + (-d3 * cos(150)) (from d3).
Calculating the values,
Σ dx = 0 + 2.01 / √2 + (-1.12 * sin(150)),
Σ dy = 3.98 + 2.01 / √2 + (-1.12 * cos(150)).
By substituting the numeric values and evaluating these equations, we can find the values of Σ dx and Σ dy.
Finally, the magnitude and direction of the resultant displacement can be found using the Pythagorean theorem and inverse tangent function as:
Magnitude: |r| = sqrt((Σ dx)^2 + (Σ dy)^2),
Direction: θ = atan(Σ dx / Σ dy).
Substituting the calculated values and evaluating, we can find the magnitude and direction of the single displacement made by the expert golfer to make the hole in one stroke.