A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 4.00 m to the north, d2 = 1.90 m northeast, and d3 = 1.00 m at θ = 30.0° west of south (figure below). Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Question

A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 4.00 m to the north, d2 = 1.90 m northeast, and d3 = 1.00 m at θ = 30.0° west of south (figure below). Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Answer 1

See previous post: Tue,1-18-11,6:56 PM.

Answer 2

To find the single displacement that an expert golfer would make to sink the ball in one stroke, we need to find the resultant displacement (R) of the three successive displacements made by the novice golfer.

Let's break down each displacement into its horizontal and vertical components:

Displacement 1 (d1): 4.00 m to the north
Horizontal component (d1x): 0 m (because it is purely vertical)
Vertical component (d1y): +4.00 m

Displacement 2 (d2): 1.90 m northeast
To find the horizontal and vertical components of a displacement given its magnitude and direction, we can use trigonometric functions. In this case, we have a right triangle with the given displacement as its hypotenuse (1.90 m) and the angle as its direction (45 degrees northeast).

Horizontal component (d2x): d2 * cos(45°)
= 1.90 m * cos(45°)
= 1.90 m * 0.7071
≈ +1.3478 m

Vertical component (d2y): d2 * sin(45°)
= 1.90 m * sin(45°)
= 1.90 m * 0.7071
≈ +1.3478 m

Displacement 3 (d3): 1.00 m at θ = 30.0° west of south
To find the horizontal and vertical components of this displacement, we need to consider the angle given relative to the positive x-axis (east).

Horizontal component (d3x): d3 * cos(180° + 30°)
= 1.00 m * cos(210°)
≈ -0.8660 m

Vertical component (d3y): d3 * sin(180° + 30°)
= 1.00 m * sin(210°)
≈ -0.5000 m

Now, let's calculate the resultant displacement (R) by summing up the horizontal and vertical components:

Rx = d1x + d2x + d3x
= 0 + 1.3478 m - 0.8660 m
≈ +0.4818 m

Ry = d1y + d2y + d3y
= 4.00 m + 1.3478 m - 0.5000 m
≈ +4.8478 m

To find the magnitude (R) and direction (θ) of the resultant displacement, we can use the Pythagorean theorem and trigonometric functions:

R = √(Rx^2 + Ry^2)
= √((+0.4818 m)^2 + (+4.8478 m)^2)
≈ 4.8577 m

θ = tan^(-1)(Ry / Rx)
= tan^(-1)(+4.8478 m / +0.4818 m)
≈ 83.4°

Therefore, an expert golfer could make the hole in a single displacement of approximately 4.8577 m at an angle of 83.4° relative to the positive x-axis (east).