As a 2.0-kg object moves from (4.4 i + 5j) m to ( 11.6 i - 2j) m, the constant resultant force acting on it is equal to (41 - 9j) N. If the speed of the object at the initial position is 4.0 m/s,
what is its kinetic energy at its final position?

v_f = 10.38 m / s

Explanation:

For this exercise we can use the relationship between work and kinetic energy

          W = ΔK

note that the two quantities are scalars

Work is defined by the relation

          W = F. Δx

the bold are vectors.  The displacement is

          Δx = r_f -r₀

          Δx = (11.6 i - 2j) - (4.4 i + 5j)

          Δx = (7.2 i - 7 j) m

          W = (4 i - 9j). (7.2 i - 7 j)

remember that the dot product

           i.i = j.j = 1

           i.j = 0

           W = 4  7.2 + 9  7

           W = 91.8 J

the initial kinetic energy is

           Ko = ½ m vo²

           Ko = ½ 2.0 4.0²

           Ko = 16 J

we substitute in the initial equation

          W = K_f - K₀

          K_f = W + K₀

          ½ m v_f² = W + K₀

          v_f² = 2 / m (W + K₀)

          v_f² = 2/2 (91.8 + 16)

           v_f = √107.8

          v_f = 10.38 m / s

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