A golfer, putting on a green, requires three strokes to "hole the ball." During the first putt, the ball rolls 5.9 m due east. For the second putt, the ball travels 2.8 m at an angle 20° north of east. The third putt is 0.50 m due north. What displacement (magnitude and direction relative to due east) would have been needed to "hole the ball" on the very first putt?
Displacement =
=sqrt{(5.9+2.8•cos20º)²+0.5²}=....
9.6
To find the displacement needed to "hole the ball" on the first putt, we need to add up the individual displacements of each putt.
First, we can break down the displacement of the second putt into horizontal and vertical components. The horizontal component is calculated by multiplying the distance by the cosine of the angle, and the vertical component is calculated by multiplying the distance by the sine of the angle.
Horizontal component of the second putt = 2.8 m * cos(20°)
Vertical component of the second putt = 2.8 m * sin(20°)
Next, we add up the total horizontal displacement by summing up the displacements of the first and second putts. The displacement to the east is positive, so we add the eastward displacement of 5.9 m to the eastward component of the second putt.
Total horizontal displacement = 5.9 m + (2.8 m * cos(20°))
Finally, to calculate the magnitude and direction of the displacement relative to due east, we can use the Pythagorean theorem to find the total displacement and trigonometry to determine the angle.
Magnitude of displacement = sqrt((Total horizontal displacement)^2 + (Vertical component of the third putt)^2)
Direction of displacement relative to due east = atan((Vertical component of the third putt) / (Total horizontal displacement))
By plugging in the values into the equations above, we can find the answers.