To determine the velocity of mass 2 after an elastic collision between two masses, we use the principles of conservation of momentum and kinetic energy. Here’s the detailed step-by-step solution:
### Step 1: Understand the Problem
We are given the following initial conditions:
- Mass 1 ([tex]\( m_1 \)[/tex]) = 281 kg
- Initial velocity of Mass 1 ([tex]\( v_{1i} \)[/tex]) = 2.82 m/s
- Mass 2 ([tex]\( m_2 \)[/tex]) = 209 kg
- Initial velocity of Mass 2 ([tex]\( v_{2i} \)[/tex]) = 1.72 m/s
We need to find the final velocity of Mass 2 ([tex]\( v_{2f} \)[/tex]) after an elastic collision.
### Step 2: Key Equations for Elastic Collisions
In an elastic collision, both momentum and kinetic energy are conserved. The final velocities of the masses after an elastic collision can be calculated using the following formulas:
For the final velocity of mass 1 ([tex]\( v_{1f} \)[/tex]):
[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]
For the final velocity of mass 2 ([tex]\( v_{2f} \)[/tex]):
[tex]\[ v_{2f} = \left( \frac{2m_1}{m_1 + m_2} \right) v_{1i} + \left( \frac{m_2 - m_1}{m_1 + m_2} \right) v_{2i} \][/tex]
### Step 3: Substitute the Given Values
First, let's substitute the given values into the formula for [tex]\( v_{2f} \)[/tex]:
[tex]\[ v_{2f} = \left( \frac{2 \times 281}{281 + 209} \right) \times 2.82 + \left( \frac{209 - 281}{281 + 209} \right) \times 1.72 \][/tex]
### Step 4: Simplify the Fractions
Calculate the denominators and numerators separately:
- [tex]\( m_1 + m_2 = 281 + 209 = 490 \)[/tex]
- [tex]\( 2 \times m_1 = 2 \times 281 = 562 \)[/tex]
- [tex]\( m_2 - m_1 = 209 - 281 = -72 \)[/tex]
### Step 5: Compute Each Term
Substitute these values back into the formula:
[tex]\[ v_{2f} = \left( \frac{562}{490} \right) \times 2.82 + \left( \frac{-72}{490} \right) \times 1.72 \][/tex]
Step-by-step calculations:
1. Calculate the first term:
[tex]\[ \frac{562}{490} \approx 1.146938775510204 \][/tex]
[tex]\[ 1.146938775510204 \times 2.82 \approx 3.2343765306122448 \][/tex]
2. Calculate the second term:
[tex]\[ \frac{-72}{490} \approx -0.146938775510204 \][/tex]
[tex]\[ -0.146938775510204 \times 1.72 \approx -0.2527438775510204 \][/tex]
### Step 6: Add the Terms
[tex]\[ v_{2f} = 3.2343765306122448 - 0.2527438775510204 \approx 2.981632653061224 \][/tex]
Therefore, the final velocity of mass 2 after the elastic collision is:
[tex]\[ v_{2f} \approx 2.982 \, \text{m/s} \][/tex]
So, the velocity of mass 2 after the collision is approximately 2.982 m/s.
