Final answer:
To calculate the pinball's total displacement, you sum up the x and y components of each vector separately and find the resultant vector's magnitude and direction, which is approximately 96.60 cm at an angle of 88.7 degrees from the horizontal.
Step-by-step explanation:
To find the total displacement of the pinball, we need to add the vectors given by their magnitude and direction. We first break down each vector into its horizontal (x) and vertical (y) components using trigonometric functions:
- The first vector is 83 cm at 90 degrees, which gives us 0 cm in the x-direction and 83 cm in the y-direction.
- The second vector is 59 cm at 147 degrees, resulting in -50.47 cm x and 44.32 cm y.
- The third vector is 69 cm at 221 degrees, resulting in -51.55 cm x and -51.55 cm y.
- The fourth vector is 45 cm at 283 degrees, resulting in 42.43 cm x and -10.79 cm y.
- The final vector is 69 cm at 27 degrees, resulting in 61.56 cm x and 31.15 cm y.
To find the resultant vector, sum the x and y components separately:
- Sum of x-components: 0 - 50.47 - 51.55 + 42.43 + 61.56 = 1.97 cm
- Sum of y-components: 83 + 44.32 - 51.55 - 10.79 + 31.15 = 96.13 cm
The magnitude of the resultant displacement vector (S) is calculated using the Pythagorean theorem:
S = √(x² + y²) = √(1.97² + 96.13²) = √(9330.56) = 96.60 cm approximately
The angle made with the horizontal (θ) is found using the tangent function:
θ = arctan(y/x) = arctan(96.13/1.97) ≈ 88.7 degrees