Question

Two students in bumper cars want to prove conservation of momentum. They collide theirbumper cars. The first bumper car has a mass of
120
kg
and was moving with a velocity of
4.0
m/s
before the collision and with a velocity of
−
2.0
m/s
after the collision. The second bumper car has amass of
90
kg
and was moving at a velocity of
−
5.0
m/s
before the collision. To prove conservationof momentum, what must the velocity of the second bumper car be after the collision?
(1 point)
2.3 m/s
2.7 m/s
1.0 m/s
3.0 m/s

Answer 1

To verify conservation of momentum in a collision, we need to equate the total momentum before the collision to the total momentum after the collision.

### Given Data:

1. **First bumper car:**
- Mass (\(m_1\)) = 120 kg
- Initial velocity (\(u_1\)) = 4.0 m/s
- Final velocity (\(v_1\)) = -2.0 m/s

2. **Second bumper car:**
- Mass (\(m_2\)) = 90 kg
- Initial velocity (\(u_2\)) = -5.0 m/s
- Final velocity (\(v_2\)) = ?

### Momentum before collision:

\[ P_{\text{initial}} = m_1 u_1 + m_2 u_2 \]

\[
P_{\text{initial}} = (120 \, \text{kg} \times 4.0 \, \text{m/s}) + (90 \, \text{kg} \times -5.0 \, \text{m/s})
\]

\[
P_{\text{initial}} = (480 \, \text{kg m/s}) + (-450 \, \text{kg m/s}) = 30 \, \text{kg m/s}
\]

### Momentum after collision:

\[ P_{\text{final}} = m_1 v_1 + m_2 v_2 \]

Substituting the known values:

\[
P_{\text{final}} = (120 \, \text{kg} \times -2.0 \, \text{m/s}) + (90 \, \text{kg} \times v_2)
\]

\[
P_{\text{final}} = -240 \, \text{kg m/s} + 90 \, \text{kg} \times v_2
\]

### Setting \(P_{\text{initial}} = P_{\text{final}}\):

\[
30 = -240 + 90 v_2
\]

Adding 240 to both sides:

\[
270 = 90 v_2
\]

Dividing both sides by 90:

\[
v_2 = \frac{270}{90} = 3.0 \, \text{m/s}
\]

### Conclusion:
The required velocity of the second bumper car after the collision to conserve momentum is **3.0 m/s**.