Vectors basketball question

Question

Three basketball players are fighting over the ball. Josh is pulling with a force of 608 N, Nick is

pulling with a force of 550 N and Mitchell is pulling with a force of 700 N. The angle between Josh

and Nick is 120o

and the angle between Nick and Mitchell is 150o

.

a) Determine the magnitude of the resultant force.

Please include clear vector diagram(s)

b) Determine the direction of the resultant force, relative to one of the forces.

Answer 1

find x and y components, then add all x components, and all y components together,

Step 1: Resolve the forces into their horizontal (x-axis) and vertical (y-axis) components.

Let's assume Josh's force is F₁, Nick's force is F₂, and Mitchell's force is F₃.

For Josh's force: Fx₁ = F₁ * cos(0°) = 608 N * cos(0°) = 608 N Fy₁ = F₁ * sin(0°) = 608 N * sin(0°) = 0 N

For Nick's force: Fx₂ = F₂ * cos(120°) = 550 N * cos(120°) = -275 N Fy₂ = F₂ * sin(120°) = 550 N * sin(120°) = 475 N

For Mitchell's force: Fx₃ = F₃ * cos(150°) = 700 N * cos(150°) = -350 N Fy₃ = F₃ * sin(150°) = 700 N * sin(150°) = 350 N

Step 2: Sum the horizontal and vertical components of the forces to find the resultant components.

Summing the horizontal components: ΣFx = Fx₁ + Fx₂ + Fx₃ = 608 N + (-275 N) + (-350 N) = -17 N

Summing the vertical components: ΣFy = Fy₁ + Fy₂ + Fy₃ = 0 N + 475 N + 350 N = 825 N

Step 3: Find the magnitude of the resultant force using the Pythagorean theorem.

Magnitude of the resultant force: |ΣF| = sqrt((ΣFx)² + (ΣFy)²) = sqrt((-17 N)² + (825 N)²) ≈ 826 N

The magnitude of the resultant force is approximately 826 N.

b) To determine the direction of the resultant force relative to one of the forces, we can use trigonometry. Let's choose Josh's force as a reference.

The angle between the resultant force and Josh's force can be found using the inverse tangent (arctan) function:

θ = arctan(ΣFy / ΣFx) = arctan(825 N / -17 N) ≈ -86.6°

Therefore, the direction of the resultant force, relative to Josh's force, is approximately -86.6°.