In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?

검증된 단계별 안내

Step 1: Analyze the graph provided. The graph shows the potential energy U(x) as a function of position x. At point A, the potential energy is approximately 2 J, and at point B, the potential energy is 0 J. The peak of the potential energy curve is around 5 J.

Step 2: Use the principle of conservation of mechanical energy. The total mechanical energy (kinetic energy + potential energy) at point B must be equal to the total mechanical energy at point A for the particle to reach point A.

Step 3: Write the equation for conservation of energy: \( K_B + U_B = K_A + U_A \), where \( K_B \) and \( K_A \) are the kinetic energies at points B and A, and \( U_B \) and \( U_A \) are the potential energies at points B and A.

Step 4: At point A, the particle must have zero kinetic energy to just reach the point (minimum speed condition). Thus, \( K_A = 0 \). Substitute \( U_B = 0 \) and \( U_A = 2 \) J into the equation: \( K_B = U_A - U_B \).

Step 5: Express \( K_B \) in terms of the particle's mass and velocity: \( K_B = \frac{1}{2}mv^2 \). Solve for the velocity \( v \) using \( v = \sqrt{\frac{2(U_A - U_B)}{m}} \), where \( m = 0.1 \) kg (converted from 100 g).

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주요 개념

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

Potential Energy

Potential energy (U) is the energy stored in an object due to its position in a force field, commonly gravitational or elastic. In the context of the graph, it represents the energy of the particle at points A and B, where the height of the curve indicates the potential energy at those positions. Understanding how potential energy changes as the particle moves is crucial for determining the minimum speed required to reach point A.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the sum of kinetic energy (KE) and potential energy (PE) at point B must equal the potential energy at point A. This relationship allows us to calculate the minimum speed needed at point B by equating the energies at both points.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. To find the minimum speed at point B, we need to determine how much kinetic energy is required to convert into potential energy as the particle moves to point A. This concept is essential for solving the problem by linking speed to energy.