Let's break down and solve the problem step by step. --- ### **Given Data** - **Car 1:** Mass \( m_1 = 2 \) kg Velocity \( v_1 = 8 \) m/s (towards Car 2) - **Car 2:** Mass \( m_2 = 1 \) kg Velocity \( v_2 = -4 \) m/s (towards Car 1, so opposite direction to Car 1) Car 2 is **10 meters from a wall** at time of collision - There is a **bumper** on Car 2 (acts like a spring with constant \( k = 10 \) N/m), but the cars lock upon collision (so the spring effect doesn't matter after collision). - **Find:** 1. \( x_1(t) \): Position of Car 1 after collision 2. \( x_2(t) \): Position of Car 2 after collision 3. Will Car 2 hit the wall? If not, is it moving toward or away from the wall? 4. At what speed should Car 1 move for the system to stay in place after collision? --- ### **Step 1: Find Velocity After Collision (Link-Up)** When the cars link up, treat as a perfectly inelastic collision: \[ (m_1 + m_2) v_f = m_1 v_1 + m_2 v_2 \] \[ (2 + 1) v_f = 2 \times 8 + 1 \times (-4) \] \[ 3 v_f = 16 - 4 = 12 \] \[ v_f = \frac{12}{3} = 4 \text{ m/s} \] So, after collision, both move together at **4 m/s** in the initial direction of Car 1 (towards the wall). --- ### **Step 2: Write \( x_1(t) \) and \( x_2(t) \)** Assume the collision happens at \( x = 0 \): - Car 1 is at \( x = 0 \) at collision. - Car 2 is at \( x = 0 \) at collision (they collide at the same point). - After collision, both move with velocity \( v_f = 4 \) m/s. So, \[ x_1(t) = 4t \] \[ x_2(t) = 4t \] Both cars move together at 4 m/s. --- ### **Step 3: Will Car 2 Hit the Wall?** Car 2 is **10 meters from the wall** at the time of collision and is moving **towards** the wall at 4 m/s after collision. **Time to hit wall:** \[ t = \frac{10~\text{m}}{4~\text{m/s}} = 2.5~\text{s} \] So, **Car 2 will hit the wall** in 2.5 seconds. --- ### **Step 4: For System to Stay in Place (v_f = 0)** Set \( v_f = 0 \): \[ (m_1 + m_2) (0) = m_1 v_1' + m_2 v_2 \] \[ 0 = m_1 v_1' + m_2 v_2 \] \[ v_1' = -\frac{m_2}{m_1} v_2 = -\frac{1}{2} \times (-4) = 2~\text{m/s} \] So, **if Car 1 is moving at 2 m/s towards Car 2 (with Car 2 still at -4 m/s), after collision the system will stay in place.** --- ## **Final Answers** ### **Positions after collision:** \[ \boxed{x_1(t) = 4t} \] \[ \boxed{x_2(t) = 4t} \] ### **Does Car 2 hit the wall?** \[ \boxed{\text{Yes, the second car will hit the wall in 2.5 seconds.}} \] ### **Speed for system to stay in place after collision:** \[ \boxed{2~\text{m/s}} \]