A golfer takes three putts to get her ball into the hole once she is on the green. Her first putt displaces the ball 71.0 ft north; the second, 6.0 ft, 45 degrees south of east; and the third, 3.0 ft, 45 degrees south of west. What is the magnitude of the displacement (in feet) that would have been needed to get the ball into the hole on the first putt?
To find the magnitude of the displacement needed for the first putt, we need to calculate the resultant displacement. Here's how we can do it step by step:
1. Convert the second and third putts' displacements into their x and y components:
- The second putt has a displacement of 6.0 ft, 45 degrees south of east. This can be broken down into x and y components as follows:
- x-component: 6.0 ft * cos(45 degrees)
- y-component: -6.0 ft * sin(45 degrees) (negative sign indicates south)
- The third putt has a displacement of 3.0 ft, 45 degrees south of west. This can be broken down into x and y components as follows:
- x-component: -3.0 ft * cos(45 degrees) (negative sign indicates west)
- y-component: -3.0 ft * sin(45 degrees) (negative sign indicates south)
2. Calculate the sum of the x and y components of all three putts:
- x-component sum: 71.0 ft + (6.0 ft * cos(45 degrees)) + (-3.0 ft * cos(45 degrees))
- y-component sum: (6.0 ft * sin(45 degrees)) + (-3.0 ft * sin(45 degrees))
3. Use the resulting x and y components to find the magnitude of the displacement:
- Magnitude of displacement = sqrt((x-component sum)^2 + (y-component sum)^2)
By following these steps, you can calculate the magnitude of the displacement needed for the first putt.