A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 4.06 m to the north, d2 = 2.05 m northeast, and d3 = 1.15 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?
magnitude m
direction ° (north of east)
To find the single displacement made by the expert golfer, we need to add up the individual displacements taken by the novice golfer.
1. Start by breaking down each displacement into its horizontal and vertical components using trigonometry. Let's analyze each displacement:
- Displacement d1: It is purely vertical, as it goes north. So, the vertical component of d1 is equal to the magnitude of d1, which is 4.06 m.
- Displacement d2: It goes northeast, which means it has both horizontal and vertical components. To find these components, we can use the fact that it forms a right triangle with sides of length 2.05 m. By applying the Pythagorean theorem, we find that the magnitude of the horizontal and vertical components of d2 is 2.05 m / √2.
- Displacement d3: It has a magnitude of 1.15 m and a direction 30.0° west of south. To find the horizontal and vertical components, we will use trigonometry. The angle given is measured from south to west, so subtracting it from 90° gives us an angle of 60.0° from the positive x-axis. The horizontal component is then 1.15 m * cos(60.0°), and the vertical component is 1.15 m * sin(60.0°).
2. Now add up all the horizontal and vertical components separately. The total horizontal component is the sum of the horizontal components of d2 and d3. Similarly, the total vertical component is the sum of the vertical components of d1, d2, and d3.
3. To find the magnitude and direction of the single displacement, use the total horizontal and vertical components calculated in the previous step. The magnitude can be found using the Pythagorean theorem, √(horizontal^2 + vertical^2). The direction can be found by taking the arctan(vertical / horizontal).
4. Finally, express the direction as the angle north of east by subtracting the calculated angle from 90°.
Calculating all the components and combining them, you will find the magnitude and direction of the single displacement made by the expert golfer.