A 51.0 kg skier skis directly down a frictionless slope angled at $10.0^{\circ}$ to the horizontal. Choose the positive direction of the $x$-axis to be downhill along the slope. A wind force with component $F_{x}$ acts on the skier. What is $F_{x}$ if the magnitude of the skier's velocity is (a) constant, (b) increasing at a rate of $1.00 \mathrm{~m} / \mathrm{s}^{2}$, and (c) increasing at a rate of $2.00 \mathrm{~m} / \mathrm{s}^{2}$.
(a) Number
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(b) Number
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(c) Number
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Question

Accepted Answer

∻Solution∻ ‖ a ⋮ For the skier to have a constant velocity, the net force acting on the skier must be zero. Therefore, the wind force component must balance the component of gravitational force acting down the slope. The gravitational force component along the slope is given by $$F_{g,x} = mg \sin( heta)$$, where $$m = 51.0 \, ext{kg}$$ and $$g = 9.81 \, ext{m/s}^2$$. Thus, the wind force is $$F_x = mg \sin(10.0^{\circ})$$. Calculating this gives: $$F_x = 51.0 imes 9.81 imes \sin(10.0^{\circ}) \approx 8.73 \, ext{N}$$. ‖ ‖ b ⋮ If the skier is accelerating at a rate of $$1.00 \, ext{m/s}^2$$, we apply Newton's second law: $$F_{net} = ma$$. The net force is the difference between the gravitational force component and the wind force: $$F_{net} = mg \sin(10.0^{\circ}) - F_x$$. Setting this equal to $$ma$$ gives us: $$F_x = mg \sin(10.0^{\circ}) - ma$$. Substituting the values, we find: $$F_x = 51.0 imes 9.81 imes \sin(10.0^{\circ}) - 51.0 imes 1.00 \approx 7.73 \, ext{N}$$. ‖ ‖ c ⋮ For an acceleration of $$2.00 \, ext{m/s}^2$$, we use the same approach as in part (b). The equation becomes: $$F_x = mg \sin(10.0^{\circ}) - ma$$. Substituting the values gives: $$F_x = 51.0 imes 9.81 imes \sin(10.0^{\circ}) - 51.0 imes 2.00 \approx 6.73 \, ext{N}$$. ‖ ∻Answer∻ ⚹ (a) 8.73 N ⚹ (b) 7.73 N ⚹ (c) 6.73 N ⚹ ∻Key Concept∻ ⚹ Newton's Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Equation: $$F = ma$$ (Net force equals mass times acceleration) ⚹ ∻Explanation∻ ⚹ The calculations for each scenario involve balancing the forces acting on the skier, using the gravitational force component and the wind force to determine the required force for constant or varying acceleration. ⚹

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