To solve this problem, we can use the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision.
-
Calculate the initial momentum of the system:
The initial momentum \( p_{\text{initial}} \) can be calculated using the formula: \[ p_{\text{initial}} = m_1 v_1 + m_2 v_2 \] where:
- \( m_1 = 1300 , \text{kg} \) (mass of the moving car)
- \( v_1 = 2.0 , \text{m/s} \) (velocity of the moving car)
- \( m_2 = 1300 , \text{kg} \) (mass of the stopped car)
- \( v_2 = 0 , \text{m/s} \) (velocity of the stopped car)
Plugging in the values: \[ p_{\text{initial}} = (1300 , \text{kg})(2.0 , \text{m/s}) + (1300 , \text{kg})(0 , \text{m/s}) = 2600 , \text{kg m/s} \]
-
Calculate the final momentum of the system after the collision:
After the collision, the two cars stick together. Let \( v_f \) be the final velocity of the combined mass after the collision. The total mass after the collision is: \[ m_{\text{total}} = m_1 + m_2 = 1300 , \text{kg} + 1300 , \text{kg} = 2600 , \text{kg} \]
The momentum after the collision \( p_{\text{final}} \) can be expressed as: \[ p_{\text{final}} = m_{\text{total}} v_f = 2600 , \text{kg} v_f \]
-
Set the initial momentum equal to the final momentum:
According to the conservation of momentum: \[ p_{\text{initial}} = p_{\text{final}} \] Substituting the known values: \[ 2600 , \text{kg m/s} = 2600 , \text{kg} v_f \]
-
Solve for \( v_f \):
\[ v_f = \frac{2600 , \text{kg m/s}}{2600 , \text{kg}} = 1.0 , \text{m/s} \]
Thus, after the collision, both cars will move forward together at a velocity of 1.0 m/s.