Tiger Woods (golfer) takes two putts to sink his ball in the hole once he is on the green. The first putt displaces the ball 6.00 m east, and the second putt displaces the ball 5.40 m south. What displacement would put the ball in the hole in one putt?
start at (0,0)
go to (6,0)
then to (6, -5.40)
so
vector is from origin to (6, -5.40)
direction angle south of east = tan^-1 (5.4 / 6) = 42 deg
42 + 90 = 132 degrees clockwise from North
magnitude = sqrt (36 + 29.2) = 8.07 meters
Well, if Tiger Woods keeps heading in this direction, he might end up in a sand trap instead of the hole! But let's calculate his displacement. We can use the Pythagorean theorem to find the overall displacement.
The displacement from the first putt is 6.00 m east, and the displacement from the second putt is 5.40 m south. So, we can form a right triangle with these sides.
Using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the sides, and c is the hypotenuse, we can calculate the displacement:
displacement = √((6.00m)^2 + (5.40m)^2)
displacement ≈ √(36.00m^2 + 29.16m^2)
displacement ≈ √65.16m^2
displacement ≈ 8.07 m
So, Tiger would need a putt of approximately 8.07 meters to sink the ball in one shot. Hopefully, he doesn't hit it too far, or it might end up on another golf course altogether!
To determine the displacement needed to put the ball in the hole in one putt, we can use vector addition.
First, let's determine the total displacement of Tiger Woods' two putts. Since the first putt displaces the ball 6.00 m east and the second putt displaces it 5.40 m south, we can represent these displacements as vectors:
First Putt: 6.00 m east (positive x-direction)
Second Putt: 5.40 m south (negative y-direction)
Now, to find the total displacement, we need to add these two vectors together. Since these vectors are at right angles to each other, we can use the Pythagorean theorem to find the magnitude of the total displacement and trigonometry to find its direction.
Using the Pythagorean theorem:
Total Displacement = √((6.00 m)^2 + (5.40 m)^2)
Calculating this, we get:
Total Displacement ≈ √(36.00 m^2 + 29.16 m^2)
Total Displacement ≈ √65.16 m^2
Total Displacement ≈ 8.08 m
Now, to find the direction of the total displacement, we can use trigonometry. The direction can be represented as an angle with respect to the positive x-axis. We can use inverse tangent (arctan) to find this angle:
θ = arctan(5.40 m / 6.00 m)
Calculating this, we get:
θ ≈ arctan(0.90)
θ ≈ 42.6°
Therefore, the displacement needed to put the ball in the hole in one putt is approximately 8.08 m at an angle of approximately 42.6° with respect to the positive x-axis.
To find the displacement that would put the ball in the hole in one putt, we need to combine the displacements of the first and second putts. The displacement of an object is a vector quantity and is determined by its magnitude (how far it travels) and its direction.
To solve this problem, we can use vector addition.
First, let's visualize the displacements on a coordinate plane. Let's assume that the starting point of the ball is the origin (0,0), and each displacement is relative to that point. The first putt displaces the ball 6.00 m east (to the right), so we can represent it as a displacement vector: (6.00, 0). The second putt displaces the ball 5.40 m south (downward), so we can represent it as a displacement vector: (0, -5.40).
To find the net displacement, we simply add the two vectors together. Adding the x-components gives us 6.00 + 0 = 6.00, and adding the y-components gives us 0 + (-5.40) = -5.40. Therefore, the net displacement is (6.00, -5.40).
Now, to find the magnitude of this net displacement, we can use the Pythagorean theorem. The magnitude of a displacement vector is given by the square root of the sum of the squares of its components. In this case, the magnitude of the net displacement is given by:
Magnitude = sqrt((6.00)^2 + (-5.40)^2) = sqrt(36.00 + 29.16) = sqrt(65.16) ≈ 8.08 m
The direction of the net displacement can be determined using trigonometry. You can find the angle θ between the positive x-axis and the net displacement vector by calculating:
θ = arctan(y-component / x-component) = arctan(-5.40 / 6.00) ≈ -40.6°
So, the displacement that would put the ball in the hole in one putt is approximately 8.08 m at an angle of approximately -40.6° (measured counterclockwise from the positive x-axis).
Note: The negative angle indicates that the displacement is in the fourth quadrant of the coordinate plane. However, in golf, the angles are typically measured clockwise from the positive x-axis, so the answer would be approximately 40.6° in that context.