To find the magnitude of the total momentum of the two balls after the collision, we first need to calculate the momentum of each ball before the collision.
Step 1: Calculate initial momentum of each ball.
The formula for momentum \( p \) is given by: \[ p = m \cdot v \] where:
- \( m \) is the mass of the object,
- \( v \) is the velocity of the object.
For the 2.0 kg ball traveling north:
- Mass (\( m_1 \)) = 2.0 kg
- Velocity (\( v_1 \)) = 4.0 m/s (north)
Momentum of the 2.0 kg ball: \[ p_1 = m_1 \cdot v_1 = 2.0 , \text{kg} \times 4.0 , \text{m/s} = 8.0 , \text{kg·m/s} , \text{(north)} \]
For the 1.0 kg ball traveling south:
- Mass (\( m_2 \)) = 1.0 kg
- Velocity (\( v_2 \)) = -8.0 m/s (south is taken as negative)
Momentum of the 1.0 kg ball: \[ p_2 = m_2 \cdot v_2 = 1.0 , \text{kg} \times (-8.0 , \text{m/s}) = -8.0 , \text{kg·m/s} , \text{(south)} \]
Step 2: Calculate total initial momentum.
Now we can find the total initial momentum (\( p_{\text{total}} \)): \[ p_{\text{total}} = p_1 + p_2 = 8.0 , \text{kg·m/s} + (-8.0 , \text{kg·m/s}) = 0 , \text{kg·m/s} \]
Step 3: Determine total momentum after collision.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Since the total initial momentum is 0 kg·m/s, the total momentum after the collision must also be 0 kg·m/s.
Conclusion: Magnitude of total momentum after collision: \[ \text{Magnitude of total momentum} = |p_{\text{total}}| = |0 , \text{kg·m/s}| = 0 , \text{kg·m/s} \]
Thus, the magnitude of the total momentum of the two balls after the collision is \( \boxed{0} , \text{kg·m/s} \).