A very slippery block of ice slides down a smooth ramp tilted at angle θ. The ice is released from rest at vertical height h above the bottom of the ramp. Find an expression for the speed of the ice at the bottom.

검증된 단계별 안내

Start by identifying the principle of conservation of energy. The total mechanical energy of the system is conserved because there is no friction or external work done. The potential energy at the top of the ramp is converted entirely into kinetic energy at the bottom.

Write the expression for gravitational potential energy at the top of the ramp: \( U = mgh \), where \( m \) is the mass of the ice, \( g \) is the acceleration due to gravity, and \( h \) is the vertical height.

Write the expression for kinetic energy at the bottom of the ramp: \( K = \frac{1}{2}mv^2 \), where \( v \) is the speed of the ice at the bottom.

Set the initial potential energy equal to the final kinetic energy using the conservation of energy principle: \( mgh = \frac{1}{2}mv^2 \). Notice that the mass \( m \) cancels out from both sides of the equation.

Solve for \( v \) by isolating it: \( v = \sqrt{2gh} \). This is the expression for the speed of the ice at the bottom of the ramp in terms of the given variables \( g \) and \( h \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy of the ice at height h is converted into kinetic energy as it slides down the ramp. This relationship allows us to equate the initial potential energy to the final kinetic energy to find the speed of the ice.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For an object at height h, the potential energy is given by the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. As the ice block descends the ramp, this potential energy decreases, converting into kinetic energy.

Kinetic Energy

Kinetic energy is the energy of an object in motion, defined by the formula KE = 1/2 mv², where m is mass and v is velocity. At the bottom of the ramp, all the potential energy has been converted into kinetic energy, allowing us to solve for the speed of the ice. Understanding this relationship is crucial for deriving the expression for the speed at the bottom of the ramp.