A 500 g wood block on a frictionless table is attached to a horizontal spring. A 50 g dart is shot into the face of the block opposite the spring, where it sticks. Afterward, the spring oscillates with a period of 1.5 s and an amplitude of 20 cm. How fast was the dart moving when it hit the block?

검증된 단계별 안내

Convert the given masses into kilograms: The mass of the wood block is 500 g = 0.5 kg, and the mass of the dart is 50 g = 0.05 kg.

Determine the total mass of the system after the dart sticks to the block: \( m_{\text{total}} = m_{\text{block}} + m_{\text{dart}} \).

Use the relationship between the period of oscillation and the spring constant: The period \( T \) is related to the spring constant \( k \) and the total mass \( m_{\text{total}} \) by the formula \( T = 2\pi \sqrt{\frac{m_{\text{total}}}{k}} \). Rearrange this equation to solve for \( k \): \( k = \frac{4\pi^2 m_{\text{total}}}{T^2} \).

Relate the amplitude and spring constant to the velocity of the block-dart system after the collision: The maximum potential energy stored in the spring is \( U = \frac{1}{2}kA^2 \), where \( A \) is the amplitude. This energy is equal to the kinetic energy of the block-dart system immediately after the collision: \( KE = \frac{1}{2}m_{\text{total}}v^2 \). Set \( U = KE \) and solve for \( v \): \( v = \sqrt{\frac{kA^2}{m_{\text{total}}}} \).

Use the principle of conservation of momentum to find the initial velocity of the dart: Before the collision, the dart has momentum \( p = m_{\text{dart}}v_{\text{dart}} \), and the block is stationary. After the collision, the total momentum is \( m_{\text{total}}v \), where \( v \) is the velocity of the block-dart system. Set the initial and final momenta equal: \( m_{\text{dart}}v_{\text{dart}} = m_{\text{total}}v \). Solve for \( v_{\text{dart}} \): \( v_{\text{dart}} = \frac{m_{\text{total}}v}{m_{\text{dart}}} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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주요 개념

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

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, when the dart collides with the wood block, the momentum of the dart is transferred to the block, allowing us to calculate the dart's initial velocity using the combined mass and the final velocity of the system.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. The period of oscillation, which is given in the question, can be used to determine the spring constant and the mass of the system, providing insights into the dynamics of the block-spring system after the collision.

Kinetic Energy and Potential Energy in Springs

In a spring system, kinetic energy (KE) and potential energy (PE) interchange during oscillation. The potential energy stored in the spring at maximum displacement (amplitude) can be equated to the kinetic energy of the dart just before the collision, allowing us to analyze the energy transfer and calculate the dart's speed using the relationship between kinetic and potential energy.