The kilogram snowboarder stands at the top of a slope that rises 27.14 meters above the valley floor. After sliding down the slope, (s)he glides up a 6 meter long jump that rises 3 meters above the valley floor. Calculate the snowboarder's speed at the top of the ramp if the coefficient of friction of the board on the snow is 0.15 and the base of the slope is 16.30 meters out from the edg
To calculate the snowboarder's speed at the top of the ramp, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of an object remains constant if no external forces act on it. In this case, we will assume that air resistance is negligible.
Let's break down the problem step by step:
1. Calculate the potential energy at the top of the slope:
Potential energy (PE) = mass * gravitational acceleration * height
PE_top = 100 kg * 9.8 m/s^2 * 27.14 m
PE_top = 26527.2 J
2. Calculate the potential energy at the top of the jump:
PE_jump = mass * gravitational acceleration * height
PE_jump = 100 kg * 9.8 m/s^2 * 3 m
PE_jump = 2940 J
3. Calculate the initial kinetic energy at the top of the ramp:
Since the snowboarder starts from rest, the initial kinetic energy is zero.
4. Calculate the work done by friction on the slope:
The work done by friction is equal to the initial potential energy minus the potential energy at the top of the ramp:
Work_friction = PE_top - PE_ramp
Work_friction = 26527.2 J - 2940 J
Work_friction = 23587.2 J
5. Calculate the distance traveled on the slope:
We can use the Pythagorean theorem to find the distance traveled on the slope (hypotenuse of a right triangle):
Distance_slope = √((height_slope)^2 + (base_slope)^2)
Distance_slope = √((27.14 m)^2 + (16.30 m)^2)
Distance_slope = √(737.2996 + 265.69)
Distance_slope = √1003.9896
Distance_slope = 31.68 m (approx.)
6. Calculate the work done against friction on the slope:
The work done against friction is given by the force of friction multiplied by the distance traveled on the slope:
Work_against_friction = force_friction * distance_slope
force_friction = coefficient_of_friction * normal_force
The normal force is equal to mass * gravitational acceleration
force_friction = 0.15 * 100 kg * 9.8 m/s^2 = 147 N (approx.)
Work_against_friction = 147 N * 31.68 m
Work_against_friction = 4652.16 J
7. Calculate the work done by the snowboarder on the jump:
The work done by the snowboarder is equal to the potential energy at the top of the jump:
Work_jump = PE_jump = 2940 J
8. Calculate the final kinetic energy at the top of the ramp:
The final kinetic energy is equal to the initial kinetic energy plus the work done by friction on the slope plus the work done by the snowboarder on the jump:
Final KE = Initial KE + Work_against_friction + Work_jump
Final KE = 0 J + 4652.16 J + 2940 J
Final KE = 7592.16 J
9. Calculate the final velocity at the top of the ramp:
Final KE = 0.5 * mass * final velocity^2
7592.16 J = 0.5 * 100 kg * final velocity^2
final velocity^2 = 7592.16 J / 50 kg
final velocity^2 = 151.84 m^2/s^2
final velocity = √151.84 m^2/s^2
final velocity = 12.32 m/s (approx.)
Therefore, the snowboarder's speed at the top of the ramp is approximately 12.32 m/s.