A 65.0 kg skier with an initial speed of 11.0 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.
To find the final speed of the skier at the top of the rise, you need to consider the conservation of mechanical energy.
The initial mechanical energy of the skier is given by the sum of his kinetic energy and potential energy:
Initial Energy = (1/2) * mass * (initial speed)^2 + mass * gravity * height
where
mass = 65.0 kg (mass of the skier)
initial speed = 11.0 m/s (initial speed of the skier)
gravity = 9.8 m/s^2 (acceleration due to gravity)
height = 2.50 m (height of the rise)
Plugging in the values, we get:
Initial Energy = (1/2) * 65.0 kg * (11.0 m/s)^2 + 65.0 kg * 9.8 m/s^2 * 2.50 m
Next, we need to determine the amount of energy lost due to friction. Friction is responsible for converting some of the skier's initial mechanical energy into thermal energy. The work done by friction is given by the formula:
Work done by friction = coefficient of friction * normal force * distance
where
coefficient of friction = 0.0800 (given)
normal force = mass * gravity (weight of the skier)
distance = height of the rise = 2.50 m (given)
Plugging in the values, we get:
Work done by friction = 0.0800 * (65.0 kg * 9.8 m/s^2) * 2.50 m
Now, subtract the work done by friction from the initial mechanical energy to get the final mechanical energy:
Final Energy = Initial Energy - Work done by friction
Finally, we can find the final speed of the skier at the top of the rise using the conservation of mechanical energy:
Final Speed = sqrt((2 * Final Energy) / mass)
Plug in the values to calculate the final speed.