Question

Two students in bumper cars want to prove conservation of momentum. They collide their bumper 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 a mass of 90 kg and was moving at a velocity of −5.0 m/s before the collision. To prove conservation of momentum, what must the velocity of the second bumper car be after the collision?(1 point)
Responses

2.3 m/s
2 point 3 meters per seconds

1.0 m/s
1 point 0 meters per seconds

3.0 m/s
3 point 0 meters per seconds

2.7 m/s

Answer 1

To prove conservation of momentum, we need to calculate the total momentum before and after the collision and ensure they are equal.

**Step 1: Calculate the total momentum before the collision.**

The momentum \( p \) is given by the formula:
\[ p = mv \]
where \( m \) is mass and \( v \) is velocity.

For the first bumper car:
\[ p_1 = m_1 v_1 = 120 \, \text{kg} \times 4.0 \, \text{m/s} = 480 \, \text{kg m/s} \]

For the second bumper car:
\[ p_2 = m_2 v_2 = 90 \, \text{kg} \times (-5.0 \, \text{m/s}) = -450 \, \text{kg m/s} \]

Now, calculate the total momentum before the collision:
\[ p_{\text{total before}} = p_1 + p_2 = 480 \, \text{kg m/s} + (-450 \, \text{kg m/s}) = 30 \, \text{kg m/s} \]

**Step 2: Calculate the total momentum after the collision.**

Let \( v_2' \) be the velocity of the second bumper car after the collision. The momentum of the second bumper car after the collision is:
\[ p_2' = m_2 v_2' = 90 \, \text{kg} \times v_2' \]

The momentum of the first bumper car after the collision is:
\[ p_1' = m_1 v_1' = 120 \, \text{kg} \times (-2.0 \, \text{m/s}) = -240 \, \text{kg m/s} \]

Now, calculate the total momentum after the collision:
\[ p_{\text{total after}} = p_1' + p_2' = -240 \, \text{kg m/s} + 90 \, \text{kg} \times v_2' \]

**Step 3: Set the total momentum before equal to the total momentum after.**
\[ 30 = -240 + 90 v_2' \]

**Step 4: Solve for \( v_2' \).**
Add 240 to both sides:
\[ 30 + 240 = 90 v_2' \]
\[ 270 = 90 v_2' \]

Divide both sides by 90:
\[ v_2' = \frac{270}{90} = 3.0 \, \text{m/s} \]

Thus, the velocity of the second bumper car after the collision must be **3.0 m/s**.