A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are d1 = 3.98 m to the north, d2 = 2.01 m northeast, and d3 = 1.12 m at 30.0° west of south (see the figure). 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 = 3.98 m to the north, d2 = 2.01 m northeast, and d3 = 1.12 m at 30.0° west of south (see the figure). Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Answer 1

d1 = (0,3.98)

d2 = (2.01/√2,2.01/√2)
d3 = (-1.12/2,-1.12√3/2)

just add them up and convert back to compass directions. If you get stuck, just say where.

Answer 2

To find the single displacement that an expert golfer could make the hole in, we need to combine the three successive displacements of the novice golfer.

We are given the following displacements:
d1 = 3.98 m to the north
d2 = 2.01 m northeast
d3 = 1.12 m at 30.0° west of south

To combine these displacements, we can use vector addition. We need to find the resultant displacement, which is the final displacement after all three displacements have been added together.

Step 1: Convert d2 and d3 into their respective horizontal and vertical components.

For d2 (2.01 m northeast):
The displacement is at an angle of 45° with the positive x-axis (northeast is a combination of north and east directions).
The horizontal component (dx2) = d2 * cos(45) = 2.01 * cos(45) = 1.42 m
The vertical component (dy2) = d2 * sin(45) = 2.01 * sin(45) = 1.42 m

For d3 (1.12 m at 30.0° west of south):
The displacement is at an angle of 30° west of south, which means it is 60° east of south.
The horizontal component (dx3) = d3 * cos(60) = 1.12 * cos(60) = 0.56 m
The vertical component (dy3) = -d3 * sin(60) = -1.12 * sin(60) = -0.97 m (negative sign because it's in the opposite direction of south)

Step 2: Add the horizontal and vertical components separately to find the resultant horizontal (Rx) and vertical (Ry) components.

Rx = d1 + dx2 + dx3 = 3.98 + 1.42 + 0.56 = 5.96 m
Ry = dy2 + dy3 = 1.42 - 0.97 = 0.45 m

Step 3: Find the magnitude (R) and direction (θ) of the resultant displacement.

R = √(Rx^2 + Ry^2) = √(5.96^2 + 0.45^2) = √35.55 = 5.96 m (rounded to two decimal places)
θ = arctan(Ry / Rx) = arctan(0.45 / 5.96) = arctan(0.075) = 4.31° (rounded to two decimal places)

Therefore, the expert golfer could make the hole in a displacement of approximately 5.96 m at an angle of 4.31° above the positive x-axis.

d1 = (0,3.98)

4.64

To find the single displacement that an expert golfer could make the hole in, we need to combine the three successive displacements of the novice golfer.