A constant force (3,0,4)N moves a body from point (-5,3,1)m to the point (4,3,3)m. If the initial velocity of the body is (6,1,1)m/s, determine the work done by the force and the angle between the force and the initial velocity of the body​

The net displacement of the body is

∆x = (4, 3, 3) m - (-5, 3, 1) m = (9, 0, 2) m

so the work done by F = (3, 0, 4) N in the direction of ∆x is

F • ∆x = (3, 0, 4) • (9, 0, 2) N•m = (27 + 0 + 8) N•m = 35 J

The angle between the force and initial velocity v₀ is θ, such that

F • v₀ = ||F|| ||v₀|| cos(θ)

(3, 0, 4) • (6, 1, 1) N•m/s = √(3² + 0² + 4²) √(6² + 1² + 1²) cos(θ) N•m/s

==>   cos(θ) = (18 + 0 + 4) / (√25 × √38) = 22/(5√38)

==>   θ = arccos(22/(5√38)) ≈ 44.5°

magnitude and direction

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