A 50.0 kg skier with an initial speed of 12.0 m/s coasts up a 2.50 m high rise as shown in Figure 6.23. Find his final speed at the top, given that the coefficient of friction between her skis and the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.)
To find the skier's final speed at the top of the rise, we need to consider the conservation of mechanical energy. The skier starts with an initial kinetic energy and gravitational potential energy and ends with a final kinetic energy at the top.
First, let's calculate the initial kinetic energy of the skier. The formula for kinetic energy is given by:
Kinetic Energy = (1/2) × mass × velocity^2
Plugging in the values, we have:
Initial Kinetic Energy = (1/2) × 50.0 kg × (12.0 m/s)^2
Next, let's calculate the gravitational potential energy of the skier at the top of the rise. The formula for gravitational potential energy is:
Gravitational Potential Energy = mass × g × height
where g is the acceleration due to gravity (9.8 m/s^2) and height is the vertical distance traveled up the rise (2.50 m).
Plugging in the values, we have:
Gravitational Potential Energy = 50.0 kg × 9.8 m/s^2 × 2.50 m
Now, let's consider the work done by friction on the skier's motion. The work done by friction is given by the formula:
Work by Friction = friction force × distance
To find the distance traveled up the incline, we need to use the hint given in the question, assuming a straight-line path. Let's denote this distance as d.
The friction force can be calculated using the formula:
Friction Force = coefficient of friction × Normal Force
The normal force is given by:
Normal Force = mass × g × cos(θ)
where θ is the angle of the incline (the angle between the incline and the horizontal). Based on the given information, the angle is not provided in the question, so I cannot provide a direct solution to the problem without more information. However, you can find the angle or use a trigonometric function to calculate the friction force if you have the necessary information.
Once you find the friction force, you can multiply it by the distance traveled up the incline (d) to find the work done by friction.
Now, let's summarize the calculations:
1. Calculate the initial kinetic energy using the skier's mass and initial velocity.
2. Calculate the gravitational potential energy at the top using the skier's mass, acceleration due to gravity, and the vertical height traveled.
3. Find the distance traveled up the incline (d) using the hint given in the question or using additional information.
4. Determine the friction force using the coefficient of friction and the normal force.
5. Multiply the friction force by the distance traveled up the incline (d) to find the work done by friction.
6. Apply the conservation of mechanical energy by equating the initial kinetic energy and gravitational potential energy to the final kinetic energy.
7. Solve for the final velocity at the top of the rise.