Bumper car \( 1(281 \mathrm{~kg}) \) moves east at \( 2.82 \mathrm{~m} / \mathrm{s} \). Bumper car \( 2(209 \mathrm{~kg}) \) moves west at \( 1.72 \mathrm{~m} / \mathrm{s} \). If the cars have an elastic collision, what would the velocity of car 2 be afterwards? \( \mathrm{v}_{2 \mathrm{f}}=[?] \mathrm{m} / \mathrm{s} \) Remember: East is positive \( (+) \); west is negative \( (-) \).

To solve this problem, we need to use the principles of conservation of momentum and kinetic energy, as it's an elastic collision. Let's denote: - \( m_1 = 281 \, \text{kg} \) (mass of car 1) - \( v_{1i} = 2.82 \, \text{m/s} \) (initial velocity of car 1, going east) - \( m_2 = 209 \, \text{kg} \) (mass of car 2) - \( v_{2i} = -1.72 \, \text{m/s} \) (initial velocity of car 2, going west) Using the conservation of momentum: \[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \] Where \( v_{1f} \) is the final velocity of car 1, and \( v_{2f} \) is the final velocity of car 2. Using the conservation of kinetic energy: \[ \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \] For elastic collisions, we can also use the relative velocity equation: \[ v_{1i} - v_{2i} = -(v_{1f} - v_{2f}) \] By solving this set of equations, we can directly find \( v_{2f} \) as follows: 1. From the conservation of momentum: \[ (281)(2.82) + (209)(-1.72) = (281)(v_{1f}) + (209)(v_{2f}) \] This simplifies down to: \[ 792.42 - 359.88 = 281 v_{1f} + 209 v_{2f} \] \[ 432.54 = 281 v_{1f} + 209 v_{2f} \tag{1} \] 2. From the relative velocity equation: \[ 2.82 - (-1.72) = -(v_{1f} - v_{2f}) \] \[ 4.54 = -v_{1f} + v_{2f} \] \[ v_{2f} = v_{1f} + 4.54 \tag{2} \] 3. Substitute equation (2) into equation (1): \[ 432.54 = 281 v_{1f} + 209(v_{1f} + 4.54) \] \[ 432.54 = 281 v_{1f} + 209 v_{1f} + 950.26 \] \[ 432.54 - 950.26 = (281 + 209)v_{1f} \] \[ -517.72 = 490 v_{1f} \implies v_{1f} = -1.057 \, \text{m/s} \] 4. Now, substitute \( v_{1f} \) back into equation (2) to find \( v_{2f} \): \[ v_{2f} = -1.057 + 4.54 \approx 3.483 \, \text{m/s} \] Thus, the final velocity of car 2 after the elastic collision is approximately: \[ \mathrm{v}_{2 \mathrm{f}} \approx 3.48 \, \mathrm{m/s} \]