To find the velocity after the collision, we can use the principle of conservation of momentum. The total momentum before the collision equals the total momentum after the collision.
The momentum before the collision is given by the sum of the momentum of both carts:
- Momentum of Cart 1: \( m_1 v_1 = 2.0 , \text{kg} \times 3.5 , \text{m/s} = 7.0 , \text{kg m/s} \)
- Momentum of Cart 2: Since it is at rest, \( m_2 v_2 = 2.0 , \text{kg} \times 0 , \text{m/s} = 0 , \text{kg m/s} \)
The total momentum before the collision is: \[ 7.0 + 0 = 7.0 , \text{kg m/s} \]
After the collision, both carts stick together, so their combined mass is: \[ m_1 + m_2 = 2.0 , \text{kg} + 2.0 , \text{kg} = 4.0 , \text{kg} \]
Let \( v_f \) be the final velocity after the collision. The total momentum after the collision is: \[ (m_1 + m_2) v_f = 4.0 , \text{kg} \times v_f \]
Setting the total momentum before the collision equal to the total momentum after the collision gives: \[ 7.0 , \text{kg m/s} = 4.0 , \text{kg} \times v_f \]
Now, solve for \( v_f \): \[ v_f = \frac{7.0 , \text{kg m/s}}{4.0 , \text{kg}} = 1.75 , \text{m/s} \]
Therefore, the velocity after the collision is: \[ \boxed{1.75} \]