How Far Must a Skier Travel to Reach 48 m/s on a 22° Slope?

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In summary, using the work-KE theorem and disregarding friction, a professional skier starting from rest on a ski slope angled 22.0° above the horizontal would need to travel a minimum distance of 310 meters to reach a speed of 48 m/s. This is determined by the equation W = 1/2 mvf2 = mgdsinθ, which agrees with the given solution. Although one student argues that cos θ should be used instead of sin θ, checking the zero angle case confirms that sin θ is the correct choice.
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Homework Statement


A professional skier starts from rest and reaches a speed of 48 m/s on a ski slope angled 22.0° above the horizontal. Using the work-KE theorem and disregarding friction, find the minimum distance along the slope the skier would have to travel in order to reach this speed?

Homework Equations


W = 1/2mvf2
W = Fd = mgd

The Attempt at a Solution



1/2 mvf2 = mgdsinθ

My solution is 310 M. Is this reasonable given the scenario? It makes sense to me but one of my students is sure it is incorrect. He says that we should be using the cos θ rather than the sin θ. The solution agrees with the key. I want to be able clearly explain the situation to him.
 
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I always check I'm choosing correctly between sin and cos by considering the zero angle case. If theta is zero here, what will the velocity be?
 

FAQ: How Far Must a Skier Travel to Reach 48 m/s on a 22° Slope?

What is work-kinetic energy?

Work-kinetic energy is the relationship between the work done on an object and the resulting change in its kinetic energy. It is a fundamental concept in physics that explains how energy is transferred and transformed in systems.

How is work-kinetic energy related to skiing?

In skiing, work-kinetic energy is used to describe the energy transfer between the skier and the snow. As the skier moves down the slope, the work done by gravity and friction results in a change in the skier's kinetic energy, which determines their speed and motion.

What is the equation for work-kinetic energy?

The equation for work-kinetic energy is W = Fd = ½mv2 - ½mv02, where W is the work done, F is the net force, d is the distance traveled, m is the mass of the object, v is the final velocity, and v0 is the initial velocity.

How can work-kinetic energy be calculated for a skier?

To calculate work-kinetic energy for a skier, we need to know the mass of the skier, the distance traveled, and the change in velocity. The force can be calculated using the skier's weight and the angle of the slope. Then, we can plug these values into the equation W = Fd = ½mv2 - ½mv02 to find the work done and the change in kinetic energy.

What factors affect work-kinetic energy in skiing?

The work-kinetic energy in skiing is affected by several factors, including the slope angle, the skier's weight, the friction between the skis and the snow, and the air resistance. These factors can influence the force and distance traveled, thus affecting the amount of work done and the change in kinetic energy.

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