A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 3.96 m to the north, d2 = 2.06 m northeast, and d3 = 1.13 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?
I got 4.88 m and 65.5 N of E.
D=3.96m[90o] + 2.06m[45o] + 1.13m[240o]
X=3.96*cos90+2.06*cos45+1.13*cos240 =0.892 m.
Y = 3.96*sin90+2.06*sin45+1.13*sin240 =
4.44 m.
tan A = Y/X = 4.44/0.892 = 4.98
A = 78.6o
D = X/cos A = 0.892/cos78.6 = 4.51 m @ 78.6o N of E.
Henry how did you know that the angles were 90, 45 and 240?
You know the angles by the "north," "northeast," and "30 degrees west of south" (180+30)
To find the single displacement of the expert golfer, we can add together the three given displacements, taking into account both their magnitudes and directions.
First, let's consider the displacement d1 = 3.96 m to the north. This displacement is purely in the north direction, so its contribution to the total displacement will be along the north direction.
Next, let's consider the displacement d2 = 2.06 m northeast. This displacement is at an angle of 45 degrees to the north-east direction. To find its components along the north and east directions, we can use trigonometry:
Component along the north direction = d2 * sin(45°)
Component along the east direction = d2 * cos(45°)
Since the expert golfer is trying to reach the hole, the northward component will positively contribute to the total north displacement, and the eastward component will positively contribute to the total east displacement.
Finally, let's consider the displacement d3 = 1.13 m at 30.0° west of south. This displacement is at an angle of 30 degrees to the west of the south direction. To find its components along the south and west directions, we can use trigonometry:
Component along the south direction = d3 * sin(30°)
Component along the west direction = d3 * cos(30°)
Since the expert golfer is trying to reach the hole, the southward component will negatively contribute to the total south displacement (since it is opposite to the desired direction), and the westward component will negatively contribute to the total west displacement (since it is opposite to the desired direction).
Now, add together all the components to find the total displacement:
Total north displacement = d1 + component_north_d2 - component_south_d3
Total east displacement = component_east_d2 - component_west_d3
To find the magnitude and direction of the total displacement, we can use the Pythagorean theorem and trigonometry:
Magnitude = sqrt((Total north displacement)^2 + (Total east displacement)^2)
Direction = arctan(Total east displacement / Total north displacement)
By performing these calculations, you should be able to obtain the answer to the problem.