The angle between the force and displacement vectors can be found using the dot product and the equation F · d = ΔKE. For a change in kinetic energy of 30.0 J and -30.0 J, the angle is approximately 59.2° in both cases.
To find the angle between the force and the displacement, we first need to calculate the magnitudes of both vectors. The magnitude of the force vector, F, is 12.0 N. The magnitude of the displacement vector, d, is the square root of [(2.00)^2 + (-4.00)^2 + (3.00)^2] m, which is approximately 5.92 m.
We can now calculate the dot product of the force and displacement vectors, which is equal to the product of their magnitudes multiplied by the cosine of the angle between them.
The dot product is given by F · d = |F| |d| cos(θ), where θ is the angle we want to find.
For part (a), where the change in kinetic energy is 30.0 J, we can rearrange the equation F · d = ΔKE to solve for cos(θ). Substituting the values we obtained earlier, we have 12.0 N × 5.92 m × cos(θ) = 30.0 J. Solving for cos(θ), we find that cos(θ) = 30.0 J / (12.0 N × 5.92 m), which is approximately 0.504. Taking the inverse cosine of 0.504, we find that θ ≈ 59.2°.
For part (b), where the change in kinetic energy is -30.0 J, we follow the same steps as before. However, in this case, we have -12.0 N × 5.92 m × cos(θ) = -30.0 J. Solving for cos(θ), we find that cos(θ) = -30.0 J / (-12.0 N × 5.92 m), which is also approximately 0.504. The inverse cosine of 0.504 gives us θ ≈ 59.2°.