\begin{tabular}{l} A girl of mass \( m_{1}=60.0 \) kilograms springs from a trampoline with an initial \\ upward velocity of \( v_{i}=8.00 \) meters per second. At height \( h=2.00 \) meters \\ above the trampoline, the girl grabs a box of mass \( m_{2}=15.0 \) kilograms. ( \\ Figure 1) \\ For this problem, use \( g=9.80 \) meters per second per second for the \\ magnitude of the acceleration due to gravity. \\ \begin{tabular}{l} Is this "collision" elastic or inelastic? \\ \hline\end{tabular} \\ \hline\end{tabular}

To determine whether the collision between the girl and the box is elastic or inelastic, we need to analyze the situation based on the definitions of elastic and inelastic collisions. 1. **Elastic Collision**: In an elastic collision, both momentum and kinetic energy are conserved. This means that after the collision, the total kinetic energy of the system remains the same as it was before the collision. 2. **Inelastic Collision**: In an inelastic collision, momentum is conserved, but kinetic energy is not. In fact, some kinetic energy is transformed into other forms of energy (like heat, sound, etc.), and the objects may stick together after the collision. ### Step 1: Analyze the Situation - The girl is moving upward with an initial velocity \( v_i = 8.00 \) m/s. - She grabs a box of mass \( m_2 = 15.0 \) kg at a height \( h = 2.00 \) m above the trampoline. ### Step 2: Determine the Nature of the Collision When the girl grabs the box, she is effectively combining her mass with the box's mass. This means that they will move together after the collision. ### Step 3: Conservation of Momentum We can calculate the momentum before and after the collision to see if kinetic energy is conserved. - **Before the Collision**: - The momentum of the girl \( p_1 = m_1 v_i \) - The box is initially at rest, so its momentum \( p_2 = 0 \) The total momentum before the collision is: \[ p_{\text{initial}} = p_1 + p_2 = m_1 v_i + 0 = m_1 v_i \] - **After the Collision**: Let \( v_f \) be the final velocity of the girl and the box together after the collision. The total mass after the collision is \( m_1 + m_2 \). The total momentum after the collision is: \[ p_{\text{final}} = (m_1 + m_2) v_f \] ### Step 4: Set Up the Momentum Conservation Equation Setting the initial momentum equal to the final momentum: \[ m_1 v_i = (m_1 + m_2) v_f \] ### Step 5: Kinetic Energy Consideration To determine if the collision is elastic or inelastic, we also need to check the kinetic energy before and after the collision. - **Initial Kinetic Energy**: \[ KE_{\text{initial}} = \frac{1}{2} m_1 v_i^2 \] - **Final Kinetic Energy**: \[ KE_{\text{final}} = \frac{1}{2} (m_1 + m_2) v_f^2 \] ### Conclusion Since the girl grabs the box and they move together after the collision, this indicates that some kinetic energy is likely transformed into other forms of energy (like potential energy or sound). Therefore, the collision is inelastic. ### Final Answer The collision between the girl and the box is **inelastic**.