To prove conservation of momentum, we need to calculate the total momentum before and after the collision and ensure they are equal.
Step 1: Calculate the total momentum before the collision.
The momentum \( p \) is given by the formula: \[ p = mv \] where \( m \) is mass and \( v \) is velocity.
For the first bumper car: \[ p_1 = m_1 v_1 = 120 , \text{kg} \times 4.0 , \text{m/s} = 480 , \text{kg m/s} \]
For the second bumper car: \[ p_2 = m_2 v_2 = 90 , \text{kg} \times (-5.0 , \text{m/s}) = -450 , \text{kg m/s} \]
Now, calculate the total momentum before the collision: \[ p_{\text{total before}} = p_1 + p_2 = 480 , \text{kg m/s} + (-450 , \text{kg m/s}) = 30 , \text{kg m/s} \]
Step 2: Calculate the total momentum after the collision.
Let \( v_2' \) be the velocity of the second bumper car after the collision. The momentum of the second bumper car after the collision is: \[ p_2' = m_2 v_2' = 90 , \text{kg} \times v_2' \]
The momentum of the first bumper car after the collision is: \[ p_1' = m_1 v_1' = 120 , \text{kg} \times (-2.0 , \text{m/s}) = -240 , \text{kg m/s} \]
Now, calculate the total momentum after the collision: \[ p_{\text{total after}} = p_1' + p_2' = -240 , \text{kg m/s} + 90 , \text{kg} \times v_2' \]
Step 3: Set the total momentum before equal to the total momentum after. \[ 30 = -240 + 90 v_2' \]
Step 4: Solve for \( v_2' \). Add 240 to both sides: \[ 30 + 240 = 90 v_2' \] \[ 270 = 90 v_2' \]
Divide both sides by 90: \[ v_2' = \frac{270}{90} = 3.0 , \text{m/s} \]
Thus, the velocity of the second bumper car after the collision must be 3.0 m/s.