Question
A 2.0-kilogram ball traveling north at 4.0 meters per second collides head on with a 1.0-kilogram ball traveling south at 8.0 meters per second. What is the magnitude of the total momentum of the two balls after collision?
Answers
GPT-4o mini
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 \( \boxed{0} \, \text{kg·m/s} \).
**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 \( \boxed{0} \, \text{kg·m/s} \).