A novice golfer on the green takes three strokes to sink the ball. The
successive
displacements are 3.80 m to the north, 1.80 m northeast, and 1.00 m at 25.0°
west of south. Starting at the same initial point, an expert golfer could make
the
hole in what single displacement?
_______m
_________ ° (north of east)
A golfer takes three putts to get the ball into the hole. The first putt displaces the ball 15 ft north, the second 9.0 ft southeast, and the third 5.0 ft southwest. What displacement was needed to get the ball into the hole on the first putt?
To find the single displacement that the expert golfer would need to make the hole, we can add the displacements of the novice golfer together.
First, let's convert the displacements into their vector components. We'll assume that north is positive y-direction and east is positive x-direction.
The first displacement of 3.80 m to the north can be written as (0 m, 3.80 m) in Cartesian coordinates.
The second displacement of 1.80 m northeast can be split into its x and y components. The angle between the northeast direction and the positive x-axis is 45 degrees. Using trigonometry, we can calculate the x and y components:
x-component = displacement * cos(angle) = 1.80 m * cos(45°) ≈ 1.27 m
y-component = displacement * sin(angle) = 1.80 m * sin(45°) ≈ 1.27 m
So, the second displacement can be written as (1.27 m, 1.27 m) in Cartesian coordinates.
The third displacement of 1.00 m at 25.0° west of south can be split into its x and y components. The angle between the south direction and the positive x-axis is 180 degrees. The angle west of south is 180° + 25° = 205°. Using trigonometry, we can calculate the x and y components:
x-component = displacement * cos(angle) = 1.00 m * cos(205°) ≈ -0.93 m (negative since it's in the west direction)
y-component = displacement * sin(angle) = 1.00 m * sin(205°) ≈ -0.34 m (negative since it's in the south direction)
So, the third displacement can be written as (-0.93 m, -0.34 m) in Cartesian coordinates.
Now, we can add the displacements together to find the final displacement:
Final x-component = sum of all x-components = 0 m + 1.27 m + (-0.93 m) ≈ 0.34 m
Final y-component = sum of all y-components = 3.80 m + 1.27 m + (-0.34 m) ≈ 4.73 m
So, the final displacement can be written as (0.34 m, 4.73 m) in Cartesian coordinates.
To find the magnitude (distance) and direction of the final displacement, we can use the Pythagorean theorem and inverse trigonometric functions:
Magnitude = sqrt((final x-component)^2 + (final y-component)^2) ≈ sqrt((0.34 m)^2 + (4.73 m)^2) ≈ 4.76 m
Direction = atan(final y-component / final x-component) ≈ atan(4.73 m / 0.34 m) ≈ 88.1° (north of east)
Therefore, the expert golfer would need to make a single displacement of approximately 4.76 m at approximately 88.1° (north of east) to make the hole.