A golfer takes three strokes to get the ball into the hole once he is on the green. The first stroke displaces the ball 12 ft. north, the second stroke 6.0 ft. southeast, and the third stroke 3.0 ft. southwest. What displacement was needed to get the ball into the hole on the first stroke?
displacement
= (0,12) + 6(cos45°, -sin45°) + 3(-cos45°, -sin45°)
= (0,12) + (6/√2, -6/√2) + (-3/√2, -3/√2)
= (3/√2 , (12√2-9)/√2 )
= appr (2.12 , 5.64)
magnitude = 6.02
angle with x-axis:
tan Ø = 5.64/2.12
Ø ≐ 69.4°
Distances and displacement
To find the displacement needed to get the ball into the hole on the first stroke, we need to calculate the resultant displacement. Displacement is a vector quantity that represents the straight-line distance and direction from the initial position to the final position.
Let's break down the given information:
First stroke: 12 ft north
Second stroke: 6.0 ft southeast (which can be divided into components of motion: 6.0 ft south and 6.0 ft east)
Third stroke: 3.0 ft southwest (which can be divided into components of motion: 3.0 ft south and 3.0 ft west)
Now, let's add up these displacements to find the resultant displacement:
North ( + ) - South ( - ) = 12 ft north - 6.0 ft south - 3.0 ft south = 12 ft - 9.0 ft south
East ( + ) - West ( - ) = 6.0 ft east - 3.0 ft west = 3.0 ft east
So, the resultant displacement for the first stroke is 12 ft - 9.0 ft south + 3.0 ft east.
To simplify, we can combine the vertical (north-south) and horizontal (east-west) components:
Vertical component: 12 ft - 9.0 ft = 3.0 ft south
Horizontal component: 3.0 ft east
Therefore, the displacement needed to get the ball into the hole on the first stroke is a combination of 3.0 ft south and 3.0 ft east.
To find the displacement needed for the first stroke, we need to add up all the individual displacements in the north direction and south direction, considering their directions.
1. The first stroke displaces the ball 12 ft. north. This gives us a displacement of +12 ft. in the north direction.
2. The second stroke displaces the ball 6.0 ft. southeast. To find the north component of this displacement, we need to use the trigonometric function called sine. The angle between the southeast direction and the north direction is 45 degrees. So, we can find the north component as follows:
North component = 6.0 ft. * sin(45 degrees) = 6.0 ft. * 0.707 = 4.242 ft.
The north component of this displacement is +4.242 ft.
3. The third stroke displaces the ball 3.0 ft. southwest. Similarly, we need to use the trigonometric function called sine to find the north component from the southwest direction. The angle between the southwest direction and the north direction is also 45 degrees.
North component = 3.0 ft. * sin(45 degrees) = 3.0 ft. * 0.707 = 2.121 ft.
The north component of this displacement is -2.121 ft. since it is in the opposite direction.
Now, we can add up all the north components to find the net north displacement.
Net north displacement = 12 ft. + 4.242 ft. - 2.121 ft. = 14.121 ft.
Therefore, the displacement needed to get the ball into the hole on the first stroke in the north direction is 14.121 ft.