Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/s. They hold fast to each other after they collide. Find the magnitude and direction of the velocity of these free-spirited otters right after they collide.

검증된 단계별 안내

Step 1: Identify the principle of conservation of momentum, which states that the total momentum of a system remains constant if no external forces act on it. In this case, the collision is frictionless, so momentum is conserved.

Step 2: Write the equation for the total momentum before and after the collision. The total momentum before the collision is the sum of the individual momenta of the two otters: \( p_{\text{total, before}} = m_1 v_1 + m_2 v_2 \), where \( m_1 \) and \( v_1 \) are the mass and velocity of the first otter, and \( m_2 \) and \( v_2 \) are the mass and velocity of the second otter.

Step 3: Substitute the given values into the equation for the total momentum before the collision. Note that the direction of motion matters: assign a positive sign to the velocity of the otter moving to the right and a negative sign to the velocity of the otter moving to the left. Thus, \( v_1 = -5.00 \, \text{m/s} \) and \( v_2 = 6.00 \, \text{m/s} \).

Step 4: After the collision, the two otters stick together and move as a single object. The total momentum after the collision is \( p_{\text{total, after}} = (m_1 + m_2) v_{\text{final}} \), where \( v_{\text{final}} \) is the velocity of the combined mass. Set \( p_{\text{total, before}} = p_{\text{total, after}} \) to solve for \( v_{\text{final}} \).

Step 5: Rearrange the equation to solve for \( v_{\text{final}} \): \( v_{\text{final}} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \). Substitute the values for \( m_1 \), \( m_2 \), \( v_1 \), and \( v_2 \) into this equation to find the magnitude and direction of the final velocity of the otters after the collision.

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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 (like a collision) is equal to the total momentum after the event. This is crucial for solving collision problems, as it allows us to calculate the final velocities of objects involved in the collision by equating the total momentum before and after.

Momentum Calculation

Momentum is defined as the product of an object's mass and its velocity (p = mv). In this scenario, we calculate the momentum of each otter before the collision by multiplying their respective masses by their velocities. The direction of the momentum is also important, as it indicates the direction of motion, which will affect the final result after the collision.

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic. Inelastic collisions are those where the objects stick together after colliding, conserving momentum but not kinetic energy. In this problem, since the otters hold fast to each other after colliding, we treat it as an inelastic collision, which simplifies the calculation of their combined velocity post-collision.