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 1 be afterwards? \[ \mathrm{v}_{1 \mathrm{f}}=[?] \mathrm{m} / \mathrm{s} \] Remember: East is positive \( (+) \); west is negative \( (-) \).

To find the final velocity of bumper car \( 1 \) after the elastic collision, we can use the principles of conservation of momentum and conservation of kinetic energy. 1. First, we assign the velocities: - Initial velocity of car \( 1 \), \( v_{1i} = +2.82 \, \text{m/s} \) (east). - Initial velocity of car \( 2 \), \( v_{2i} = -1.72 \, \text{m/s} \) (west). 2. The total momentum before the collision can be calculated as: \[ p_{initial} = m_1 v_{1i} + m_2 v_{2i} = (281 \, \text{kg})(2.82 \, \text{m/s}) + (209 \, \text{kg})(-1.72 \, \text{m/s}) \] 3. The total kinetic energy before the collision is: \[ KE_{initial} = \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 \] 4. After the collision, we assume \( v_{1f} \) and \( v_{2f} \) are the final velocities of cars \( 1 \) and \( 2 \). 5. We can set up the equations based on the conservation laws: \[ m_{1} v_{1i} + m_{2} v_{2i} = m_{1} v_{1f} + m_{2} v_{2f} \] \[ \frac{1}{2} m_{1} v_{1i}^2 + \frac{1}{2} m_{2} v_{2i}^2 = \frac{1}{2} m_{1} v_{1f}^2 + \frac{1}{2} m_{2} v_{2f}^2 \] 6. Solving these equations gives us the final velocities. However, the final velocities after an elastic collision can also be calculated using a derived formula for elastic collisions: \[ v_{1f} = \frac{(m_{1} - m_{2}) v_{1i} + 2 m_{2} v_{2i}}{m_{1} + m_{2}} \] Substituting the values into this formula: \[ v_{1f} = \frac{(281 - 209)(2.82) + 2 \cdot 209(-1.72)}{281 + 209} \] \[ v_{1f} = \frac{72(2.82) - 718.96}{490} \] Calculating this gives \( v_{1f} \). After performing these calculations, we find that: \[ v_{1f} \approx 0.06 \, \text{m/s} \] Thus, the velocity of car \( 1 \) after the collision is: \[ \mathrm{v}_{1 \mathrm{f}} \approx 0.06 \, \mathrm{m/s} \]