A golfer takes three putts to get the ball into the hole. The first putt displaces the ball 3.32 m north, the second 1.63 m southeast, and the third 0.703 m southwest. What are (a) the magnitude and (b) the angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east?
this is just practice converting to and from polar to rectangular coordinates. Convert each putt, add them up, and convert back:
3.32 @ N = <0,3.32>
1.63 @ SE = <1.153,-1.153>
0.703 @ SW = <-0.497,-0.497>
add them up and you have <0.656,1.670>
and that is
1.79 @ E 67.7° N
To find the magnitude and angle between the displacement needed to get the ball into the hole in one putt and the direction due east, we first need to determine the total displacement of the golfer.
We'll break down each of the putts given:
1. The first putt displaces the ball 3.32 m north. This means there is no eastward displacement as it only moves in the north direction.
2. The second putt displaces the ball 1.63 m southeast. To simplify this, we can break it down into its northward and eastward components. Using the right-angled triangle formed by the southeast direction, we can see that the northward displacement is given by:
north displacement = 1.63 m * sin(45°) = 1.15 m (rounded to two decimal places)
And the eastward displacement is given by:
east displacement = 1.63 m * cos(45°) = 1.15 m (rounded to two decimal places)
3. The third putt displaces the ball 0.703 m southwest. Again, we'll break it down into its northward and westward components. Using the right-angled triangle formed by the southwest direction, we can calculate the northward displacement:
north displacement = 0.703 m * sin(45°) = 0.50 m (rounded to two decimal places)
And the westward displacement is given by:
west displacement = 0.703 m * cos(45°) = 0.50 m (rounded to two decimal places)
Now, let's calculate the total displacement by adding up the northward and eastward displacements and subtracting the westward displacement:
total north displacement = 3.32 m + 1.15 m + 0.50 m = 4.97 m (rounded to two decimal places)
total east displacement = 1.15 m - 0.50 m = 0.65 m (rounded to two decimal places)
To find the magnitude of the total displacement, we can use the Pythagorean theorem:
magnitude = √(total north displacement^2 + total east displacement^2)
= √(4.97^2 + 0.65^2)
= √(24.70 + 0.42)
= √25.12
= 5.01 m (rounded to two decimal places)
Finally, to find the angle between the total displacement and the direction due east, we can use the inverse tangent function:
angle = arctan(total north displacement / total east displacement)
= arctan(4.97 m / 0.65 m)
= arctan(7.65)
Using a calculator, we find that the angle is approximately 81.54° (rounded to two decimal places).
Therefore, the answers are:
(a) The magnitude of the displacement is 5.01 m.
(b) The angle between the direction of the displacement needed to get the ball into the hole in just one putt and the direction due east is approximately 81.54°.