A novice golfer on the green takes three strokes to sink the ball. The successive displacements are 4.00m north, 2.00m northeast, and 1.00m at 30.0 degrees west of south starting at the same initial point, an golfer could make the hole in what single displacement?
X = hor. = 2*cos(45) + 1*cos(240),
X = 1.414 + (-0.5) = 0.91m.
Y = ver. = 2*sin(45) + 4 + 1*sin(240),
Y = 1.414 + 4 + (-0.8660) = 4.55m.
tanA = Y/X = 4.55 / 0.91 = 5.0,
A = 78.7 deg.
D = X / cosA = 0.91/cos(78.7)= 4.64m.
To find the single displacement needed for the golfer to make the hole, we need to add up all the individual displacements.
First, let's break down the given displacements:
1. The first displacement is 4.00m north. This means the golfer moves straight north for 4.00m.
2. The second displacement is 2.00m northeast. This means the golfer moves 2.00m in both the north and east directions. The northeast direction is a combination of both north and east directions, forming a 45-degree angle.
3. The third displacement is 1.00m at 30.0 degrees west of south. This means the golfer moves 1.00m in the south and west directions. The direction given is 30 degrees west of south.
To add these displacements, we need to resolve them into their north/south and east/west components.
Let's calculate the components for each displacement:
1. The first displacement is already in the north direction, so it has a north component of 4.00m and east component of 0.00m.
2. To find the components of the second displacement, we can use trigonometry. The given displacement forms a 45-degree angle with both the north and east directions. So, the north and east components will be the same. We can use the Pythagorean theorem to find these components:
North component = cos(45°) * 2.00m = 1.41m
East component = sin(45°) * 2.00m = 1.41m
3. To find the components of the third displacement, we again use trigonometry. The given direction is 30 degrees west of south. This means it forms a 60-degree angle with the south direction. The south component will be the same as the magnitude of the displacement since it's in the south direction. The west component can be found using trigonometry:
South component = 1.00m
West component = cos(60°) * 1.00m = 0.50m
Now that we have the components, we can add them up to find the total displacement:
North component = 4.00m + 1.41m + 0.00m = 5.41m
East component = 0.00m + 1.41m + (-0.50m) = 0.91m
Using these components, we can find the magnitude and direction of the total displacement using the Pythagorean theorem and trigonometry:
Magnitude of total displacement = sqrt(5.41m^2 + 0.91m^2) = 5.47m
Direction of total displacement = atan(0.91m / 5.41m) = 9.6 degrees east of north
Therefore, the golfer could make the hole with a single displacement of approximately 5.47m at 9.6 degrees east of north.