A - A golfer takes two putts to sink his ball in the hole once he is on the green. The first putt displaces the ball 8.10 m east, and the second putt displaces it 1.99 m south. What displacement would put the ball in the hole in one putt? Answer in units of m.
-I got the answer for this as 8.3m
B - What is the direction (measured from due east, with counterclockwise positive) of the displacement? Answer in units of ◦.
- I cannot figure out how to solve for the direction of the displacement.
8.34 m ... 3 significant figures
the 1st putt is due east
the two putts form a right triangle (that you used to find the displacement)
the tangent of the angle between the 1st putt and the displacement is ... 1.99/8.10
.. this is the direction angle
... it is clockwise from east, so it will be negative
To solve for the direction of the displacement, we can use trigonometry.
First, let's break down the displacements into their respective x and y components. The first putt displaces the ball 8.10 m east, so its x-component is +8.10 m. The second putt displaces the ball 1.99 m south, so its y-component is -1.99 m.
Next, we can use these components to find the displacement vector. The displacement vector can be found by adding the x and y components together:
Displacement vector = (8.10 m) i + (-1.99 m) j,
where i represents the unit vector in the east direction and j represents the unit vector in the north direction.
Now, let's find the magnitude and direction of the displacement vector.
1. Magnitude: The magnitude of the displacement vector can be found using the Pythagorean theorem. The magnitude is given by:
Magnitude of displacement vector = √(x^2 + y^2)
= √((8.10 m)^2 + (-1.99 m)^2)
≈ 8.32 m
Thus, the magnitude of the displacement vector is approximately 8.32 m.
2. Direction: The direction of the displacement vector can be found using inverse trigonometric functions. The direction (measured from due east, with counterclockwise positive) is given by:
Direction = arc tan(y/x)
Direction = arc tan((-1.99 m) / (8.10 m))
Direction ≈ -13.6°
Therefore, the direction of the displacement vector is approximately -13.6°.
To summarize:
A. The displacement that would put the ball in the hole in one putt is approximately 8.32 m.
B. The direction (measured from due east, with counterclockwise positive) of the displacement is approximately -13.6°.
To find the displacement that would put the ball in the hole in one putt, you can use vector addition. You need to add the eastward displacement of 8.10 m to the southward displacement of 1.99 m.
To solve for the magnitude of the displacement, you can use the Pythagorean theorem:
Displacement magnitude = √(8.10^2 + 1.99^2)
= √(65.61 + 3.96)
= √69.57
≈ 8.34 m
So, the displacement that would put the ball in the hole in one putt is approximately 8.34 m.
Now, to solve for the direction of the displacement, you can use trigonometry. Since the first putt displaced the ball eastward and the second putt displaced the ball southward, you can consider the displacement as a right triangle with the eastward displacement as the adjacent side and the southward displacement as the opposite side.
The direction of the displacement (measured from due east, counterclockwise positive) can be found using the inverse tangent function (arctan).
Direction = arctan(1.99 m / 8.10 m)
Using a calculator, you can find the inverse tangent of the ratio of the southward displacement to the eastward displacement:
Direction ≈ 13.45°
Therefore, the direction of the displacement is approximately 13.45° (measured counterclockwise from due east).