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 $$ (\mathrm{t}) $$; west is negative $$ (-) $$.

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To determine the final velocity of bumper car $$ 2 $$ after an elastic collision, we'll use the principles of conservation of momentum and kinetic energy. Here's a step-by-step breakdown:

### Given:
- **Mass of Car 1 ($$ m_1 $$)**: $$ 281 \, \text{kg} $$
- **Initial Velocity of Car 1 ($$ v_{1i} $$)**: $$ 2.82 \, \text{m/s} $$ (East, positive)
- **Mass of Car 2 ($$ m_2 $$)**: $$ 209 \, \text{kg} $$
- **Initial Velocity of Car 2 ($$ v_{2i} $$)**: $$ -1.72 \, \text{m/s} $$ (West, negative)

### Formula for Final Velocity in an Elastic Collision:
The final velocity ($$ v_{2f} $$) of Car 2 after an elastic collision is given by:

$$
v_{2f} = \frac{2m_1 v_{1i} + (m_2 - m_1) v_{2i}}{m_1 + m_2}
$$

### Plugging in the Values:
1. **Calculate the Numerator:**
   $$
   2m_1 v_{1i} = 2 \times 281 \times 2.82 = 1584.84
   $$
   $$
   (m_2 - m_1) v_{2i} = (209 - 281) \times (-1.72) = (-72) \times (-1.72) = 123.84
   $$
   $$
   \text{Total Numerator} = 1584.84 + 123.84 = 1708.68
   $$

2. **Calculate the Denominator:**
   $$
   m_1 + m_2 = 281 + 209 = 490
   $$

3. **Compute $$ v_{2f} $$:**
   $$
   v_{2f} = \frac{1708.68}{490} \approx 3.487 \, \text{m/s}
   $$

### Conclusion:
After simplifying, the final velocity of **Bumper Car 2** is approximately:

$$
\mathrm{v}_{2f} \approx +3.49 \, \mathrm{m/s} \quad \text{(East)}
$$

**Answer:** After the collision, car 2 moves east with velocity 3.49  m s⁻¹.
After the collision, Bumper Car 2 moves east at approximately 3.49 m/s.

Bumper car moves east at .
Bumper car ) moves west at .
If the cars have an elastic collision,
what would the velocity of car 2 be
afterwards?

Remember: East is positive ; west is negative .

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Extra Insights

Step 1: Conservation of Momentum

Step 2: Conservation of Kinetic Energy

Step 3: Solving for Final Velocities

Answer:

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Bumper car moves east at . Bumper car ) moves west at . If the cars have an elastic collision, what would the velocity of car 2 be afterwards? Remember: East is positive ; west is negative .