A gun fires a bullet vertically into a 1.40-kg block of wood at rest on a thin horizontal sheet, Fig. 9–54. If the bullet has a mass of 15.0 g and a speed of 230 m/s, how high will the block rise into the air after the bullet becomes embedded in it?

검증된 단계별 안내

Convert the mass of the bullet from grams to kilograms. Since 1 g = 0.001 kg, the mass of the bullet is \( m_b = 15.0 \times 10^{-3} \, \text{kg} \).

Apply the principle of conservation of momentum to find the velocity of the block and bullet system immediately after the collision. The total momentum before the collision is \( p_{\text{initial}} = m_b v_b \), where \( v_b = 230 \, \text{m/s} \) is the velocity of the bullet. After the collision, the momentum is \( p_{\text{final}} = (m_b + m_w) v_f \), where \( m_w = 1.40 \, \text{kg} \) is the mass of the block and \( v_f \) is the final velocity of the combined system. Set \( p_{\text{initial}} = p_{\text{final}} \) and solve for \( v_f \).

Use the final velocity \( v_f \) of the block and bullet system as the initial velocity for vertical motion. Apply the conservation of energy principle to determine the maximum height \( h \) the system will reach. The initial kinetic energy \( KE = \frac{1}{2} (m_b + m_w) v_f^2 \) is converted entirely into gravitational potential energy \( PE = (m_b + m_w) g h \), where \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity. Set \( KE = PE \) and solve for \( h \).

Substitute the known values for \( m_b \), \( m_w \), \( g \), and \( v_f \) into the equation derived in the previous step to calculate \( h \).

Verify the units in your calculations to ensure consistency (e.g., mass in kilograms, velocity in meters per second, and height in meters). This step ensures the solution is dimensionally correct.

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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 bullet embeds itself in the block, we can apply this principle to find the combined velocity of the block and bullet immediately after the collision.

Kinetic Energy and Potential Energy

Kinetic energy is the energy of an object due to its motion, while potential energy is the energy stored in an object due to its position in a gravitational field. After the bullet embeds in the block, the kinetic energy of the block-bullet system will convert into potential energy as it rises, allowing us to calculate the maximum height reached using energy conservation principles.

A gun fires a bullet vertically into a 1.40-kg block of wood at rest on a thin horizontal sheet, Fig. 9–54. If the bullet has a mass of 15.0 g and a speed of 230 m/s, how high will the block rise into the air after the bullet becomes embedded in it?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Conservation of Momentum

Kinetic Energy and Potential Energy

Gravitational Potential Energy