A 1.0 kg mass that can move along the x-axis experiences the potential energy U = (x²−x) J, where x is in m. The mass has velocity v_x= 3.0 m/s at position x = 1.0 m. At what position has it slowed to 1.0 m/s?

검증된 단계별 안내

Step 1: Understand the problem. The mass is moving along the x-axis under the influence of a potential energy function U(x) = x² - x (in joules). The goal is to find the position x where the velocity of the mass slows down to 1.0 m/s, given that its initial velocity is 3.0 m/s at x = 1.0 m.

Step 2: Use the principle of conservation of mechanical energy. The total mechanical energy E is the sum of the kinetic energy K and the potential energy U. Write the total energy at the initial position x = 1.0 m: E = K₁ + U₁, where K₁ = (1/2)mvₓ² and U₁ = U(x).

Step 3: Write the total energy at the unknown position x where the velocity is 1.0 m/s. At this position, the total energy is E = K₂ + U₂, where K₂ = (1/2)mvₓ² and U₂ = U(x).

Step 4: Set the total energy at the initial position equal to the total energy at the unknown position, since energy is conserved. This gives the equation: (1/2)m(vₓ₁²) + U(x₁) = (1/2)m(vₓ₂²) + U(x₂). Substitute the given values for m, vₓ₁, vₓ₂, and U(x) to form the equation.

Step 5: Solve the equation for x₂. Substitute U(x) = x² - x into the equation and simplify. Rearrange the terms to isolate x₂, and solve the resulting quadratic equation to find the position where the velocity is 1.0 m/s.

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주요 개념

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

Potential Energy

Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. In this case, the potential energy U is given as a function of position x, indicating how the energy changes as the mass moves along the x-axis. Understanding how potential energy varies with position is crucial for analyzing the motion and energy transformations of the mass.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. As the mass moves and its speed changes, its kinetic energy will also change. This concept is essential for determining how the mass's speed affects its total mechanical energy when combined with potential energy.

Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in a closed system with no non-conservative forces (like friction), the total mechanical energy (kinetic plus potential) remains constant. This principle allows us to relate the potential energy and kinetic energy at different positions, enabling us to find the position where the mass slows down to a specific speed by equating the total energy at two points.