Question

A snowboarder glides down a 50-m-long, 15° hill. She then glides horizontally for 10 m before reaching a 25° upward slope. Assume the snow is frictionless. How far can she travel up the 25° slope?

Accepted Answer

Identify the key principles: This problem involves energy conservation. Since the snow is frictionless, mechanical energy is conserved. The snowboarder's initial potential energy is converted into kinetic energy as she descends the hill, and then that kinetic energy is partially converted back into potential energy as she ascends the 25-degree slope. Calculate the snowboarder's speed at the bottom of the 15-degree hill: Use the conservation of energy principle. The initial potential energy at the top of the hill is $$ U = m g h $$, where $$ h  is $$the vertical height of the hill. The vertical height can be found using $$ h = L \sin(	heta) $$, where $$ L = 50 \ 	ext{m} $$ and $$ 	heta = 15^\circ . $$The kinetic energy at the bottom is $$ K = \frac{1}{2} m v^2 . $$Set $$ U = K  to $$solve for $$ v . $$Determine the snowboarder's speed after gliding horizontally: Since the horizontal section is frictionless, no energy is lost, and the snowboarder maintains the same speed $$ v  as at $$the bottom of the hill. Set up the energy conservation equation for the 25-degree slope: As the snowboarder travels up the slope, her kinetic energy is converted back into potential energy. The potential energy at the highest point on the slope is $$ U = m g h $$, where $$ h  is $$the vertical height she reaches. The vertical height can be related to the distance traveled $$ d $$ along the slope using $$ h = d \sin(	heta) $$, where $$ 	heta = 25^\circ . $$Use the equation $$ \frac{1}{2} m v^2 = m g h  to $$solve for $$ d . $$Simplify and solve for $$ d $$: Cancel out the mass $$ m $$ from the equation, substitute $$ h = d \sin(	heta) $$, and rearrange to find $$ d . $$The final expression will depend on the initial speed $$ v $$ calculated earlier and the angle $$ 	heta .$$

A snowboarder glides down a 50-m-long, 15° hill. She then glides horizontally for 10 m before reaching a 25° upward slope. Assume the snow is frictionless. How far can she travel up the 25° slope?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Conservation of Energy

Gravitational Potential Energy

Kinematics of Inclined Planes