The butterfly takes a vertical perpendicular path equivalent to 9m and travels a horizontal distance of 17m. The net path between the two is equivalent to that of the hypotenuse, so we will apply the Pythagorean theorem.
Therefore the magnitude of the butterfly's displacement is 9m
According to the work-energy theorem, the change in the kinetic energy of the combined mass of the child and the sled, is equal to the total work done on the object by external forces.
The external forces capable to do work on the combination of child +sled, are the friction force (opposing to the displacement), and the component of the weight parallel to the slide.
As this last work is just equal to the change in the gravitational potential energy (with opposite sign) , we can write the following equation:
ΔK, is the change in kinetic energy, as follows:
ΔU, is the change in the gravitational potential energy.
If we choose as our zero reference level, the bottom of the slope, the change in gravitational potential energy will be as follows:
Finally, the work done for non-conservative forces, is the work done by the friction force, along the slope, as follows:
Replacing (2), (3), and (4) in (1), simplifying common terms, and rearranging, we have:
Replacing by the givens and the knowns, we can solve for sin θ, as follows: ⇒ θ = sin⁻¹ (0.236) = 13.7º
