An object moving in the $xy$-plane with an initial kinetic energy of $15 \mathrm{~J}$ is displaced b

Question

An object moving in the $xy$-plane with an initial kinetic energy of $15 \mathrm{~J}$ is displaced by $\vec{S}=(2 \hat{i} - \hat{j}) \mathrm{~m}$, under the continuous action of a force $\vec{F}=(4 \hat{i} + 5 \hat{j}) \mathrm{~N}$. The kinetic energy of the object at the end of this displacement is

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(b) : Given the force vector $\vec{F} = (4\hat{i} + 5\hat{j}) \mathrm{~N}$ and the displacement vector $\vec{S} = (2\hat{i} - \hat{j}) \mathrm{~m}$. The work done by the force is $W = \vec{F} \cdot \vec{S} = (4)(2) + (5)(-1) = 8 - 5 = 3 \mathrm{~J}$. According to the work-energy theorem, the total work done is equal to the change in kinetic energy: $W = K_f - K_i$. Substituting the known values: $3 = K_f - 15 \implies K_f = 18 \mathrm{~J}$.