Bumper car $$ 1(281 \mathrm{~kg}) $$ moves east at $$ 2.82 \mathrm{~m} / \mathrm{s} $$. Bumper car $$ 2(209 \mathrm{~kg}) $$ moves east 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 $$ (-) $$.

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To determine the velocity of bumper car $$ 1 $$ after an elastic collision with bumper car $$ 2 $$, we can use the principles of conservation of momentum and conservation of kinetic energy.

**Given:**
- Mass of car $$ 1 $$, $$ m_1 = 281 \, \text{kg} $$
- Initial velocity of car $$ 1 $$, $$ v_{1i} = 2.82 \, \text{m/s} $$ (east)
- Mass of car $$ 2 $$, $$ m_2 = 209 \, \text{kg} $$
- Initial velocity of car $$ 2 $$, $$ v_{2i} = 1.72 \, \text{m/s} $$ (east)

**Elastic Collision Formula:**
For a one-dimensional elastic collision, the final velocity of car $$ 1 $$ ($$ v_{1F} $$) can be calculated using the formula:

$$
v_{1F} = \frac{(m_1 - m_2)v_{1i} + 2m_2v_{2i}}{m_1 + m_2}
$$

**Plugging in the Values:**
$$
v_{1F} = \frac{(281 \, \text{kg} - 209 \, \text{kg}) \times 2.82 \, \text{m/s} + 2 \times 209 \, \text{kg} \times 1.72 \, \text{m/s}}{281 \, \text{kg} + 209 \, \text{kg}}
$$

$$
v_{1F} = \frac{72 \, \text{kg} \times 2.82 \, \text{m/s} + 418 \, \text{kg} \times 1.72 \, \text{m/s}}{490 \, \text{kg}}
$$

$$
v_{1F} = \frac{203.04 \, \text{kg} \cdot \text{m/s} + 718.96 \, \text{kg} \cdot \text{m/s}}{490 \, \text{kg}}
$$

$$
v_{1F} = \frac{922 \, \text{kg} \cdot \text{m/s}}{490 \, \text{kg}} \approx 1.88 \, \text{m/s}
$$

**Direction:**
Since the calculation yields a positive value and east is defined as positive, the direction remains east.

**Final Answer:**
$$
\mathrm{v}_{1\mathrm{F}} \approx 1.88 \, \mathrm{m/s}
$$
After the elastic collision, bumper car 1 moves east at approximately $$ 1.88 \, \text{m/s} $$.

Bumper car moves east at .
Bumper car moves east at .
If the cars have an elastic collision,
what would the velocity of car 1 be
afterwards?

Remember East is positive : west is negative .

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Bumper car moves east at . Bumper car moves east at . If the cars have an elastic collision, what would the velocity of car 1 be afterwards? Remember East is positive : west is negative .