To find the final velocity of bumper car 1 after an elastic collision with bumper car 2, we can use the given formula:
[tex]\[ v_{1f} = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) v_{1i} + \left( \frac{2m_2}{m_1 + m_2} \right) v_{2i} \][/tex]
Where:
- [tex]\( m_1 \)[/tex] is the mass of car 1,
- [tex]\( m_2 \)[/tex] is the mass of car 2,
- [tex]\( v_{1i} \)[/tex] is the initial velocity of car 1,
- [tex]\( v_{2i} \)[/tex] is the initial velocity of car 2,
- [tex]\( v_{1f} \)[/tex] is the final velocity of car 1.
Given the values:
- [tex]\( m_1 = 296 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 222 \, \text{kg} \)[/tex]
- [tex]\( v_{1i} = 3.74 \, \text{m/s} \)[/tex]
- [tex]\( v_{2i} = 1.85 \, \text{m/s} \)[/tex]
Let's substitute these values into the formula and solve for [tex]\( v_{1f} \)[/tex]:
1. Calculate the first term:
[tex]\[ \left( \frac{m_1 - m_2}{m_1 + m_2} \right) v_{1i} \][/tex]
Substitute the given values:
[tex]\[ \left( \frac{296 - 222}{296 + 222} \right) 3.74 \][/tex]
[tex]\[ \left( \frac{74}{518} \right) 3.74 \][/tex]
2. Calculate the fraction:
[tex]\[ \frac{74}{518} \approx 0.1429 \][/tex]
3. Multiply by [tex]\( v_{1i} \)[/tex]:
[tex]\[ 0.1429 \times 3.74 \approx 0.5345 \][/tex]
4. Calculate the second term:
[tex]\[ \left( \frac{2m_2}{m_1 + m_2} \right) v_{2i} \][/tex]
Substitute the given values:
[tex]\[ \left( \frac{2 \times 222}{296 + 222} \right) 1.85 \][/tex]
[tex]\[ \left( \frac{444}{518} \right) 1.85 \][/tex]
5. Calculate the fraction:
[tex]\[ \frac{444}{518} \approx 0.8571 \][/tex]
6. Multiply by [tex]\( v_{2i} \)[/tex]:
[tex]\[ 0.8571 \times 1.85 \approx 1.5866 \][/tex]
7. Sum the two terms to find [tex]\( v_{1f} \)[/tex]:
[tex]\[ v_{1f} = 0.5345 + 1.5866 \][/tex]
[tex]\[ v_{1f} \approx 2.12 \, \text{m/s} \][/tex]
Thus, the final velocity of car 1 is approximately [tex]\( 2.12 \, \text{m/s} \)[/tex].
