To prove the conservation of momentum, we can use the principle that the total momentum before the collision must equal the total momentum after the collision.
The formula for momentum (p) is:
\[ p = m \cdot v \]
Where:
- \( m \) is the mass
- \( v \) is the velocity
Let's denote the variables:
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Mass of the first bumper car, \( m_1 = 120 , \text{kg} \)
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Initial velocity of the first bumper car, \( v_{1i} = 4.0 , \text{m/s} \)
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Final velocity of the first bumper car, \( v_{1f} = -2.0 , \text{m/s} \)
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Mass of the second bumper car, \( m_2 = 90 , \text{kg} \)
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Initial velocity of the second bumper car, \( v_{2i} = -5.0 , \text{m/s} \)
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Final velocity of the second bumper car, \( v_{2f} \) (what we want to find)
Step 1: Calculate the total momentum before the collision
\[ \text{Total momentum before} = p_{1i} + p_{2i} \]
\[ p_{1i} = m_1 \cdot v_{1i} = 120 , \text{kg} \cdot 4.0 , \text{m/s} = 480 , \text{kg m/s} \] \[ p_{2i} = m_2 \cdot v_{2i} = 90 , \text{kg} \cdot (-5.0 , \text{m/s}) = -450 , \text{kg m/s} \]
Total momentum before: \[ \text{Total momentum before} = 480 , \text{kg m/s} - 450 , \text{kg m/s} = 30 , \text{kg m/s} \]
Step 2: Calculate the total momentum after the collision
\[ \text{Total momentum after} = p_{1f} + p_{2f} \] \[ p_{1f} = m_1 \cdot v_{1f} = 120 , \text{kg} \cdot (-2.0 , \text{m/s}) = -240 , \text{kg m/s} \] \[ p_{2f} = m_2 \cdot v_{2f} = 90 , \text{kg} \cdot v_{2f} \]
Step 3: Set total momentum before equal to total momentum after
\[ 30 , \text{kg m/s} = -240 , \text{kg m/s} + 90 , \text{kg} \cdot v_{2f} \]
Step 4: Solve for \( v_{2f} \)
Rearranging gives: \[ 90 , \text{kg} \cdot v_{2f} = 30 , \text{kg m/s} + 240 , \text{kg m/s} \] \[ 90 , \text{kg} \cdot v_{2f} = 270 , \text{kg m/s} \] \[ v_{2f} = \frac{270 , \text{kg m/s}}{90 , \text{kg}} = 3.0 , \text{m/s} \]
Thus, the velocity of the second bumper car after the collision must be 3.0 m/s.
Final answer: 3.0 m/s (response 3).