A 3.0-kg cart moving to the right with a speed of 1.0 m/s has a head-on collision with a 5.0-kg cart that is initially moving to the left with a speed of 2.0 m/s. After the collision, the 3.0-kg cart is moving to the left with a speed of 1.0 m/s. What is the final velocity of the 5.0-kg cart?

Options

A. zero m/s
B. 0.80 m/s to the right
C. 0.80 m/s to the left
D. 2.0 m/s to the right

Answer

To solve the problem of the collision between the two carts, we will use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and initial velocities
- Mass of cart 1 (m1) = 3.0 kg, initial velocity (u1) = +1.0 m/s (to the right)
- Mass of cart 2 (m2) = 5.0 kg, initial velocity (u2) = -2.0 m/s (to the left)

### Step 2: Write down the initial momentum
The total initial momentum (p_initial) of the system can be calculated as:
\[ p_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot u_2 \]
Substituting the values:
\[ p_{\text{initial}} = (3.0 \, \text{kg} \cdot 1.0 \, \text{m/s}) + (5.0 \, \text{kg} \cdot -2.0 \, \text{m/s}) \]
\[ p_{\text{initial}} = 3.0 \, \text{kg m/s} - 10.0 \, \text{kg m/s} = -7.0 \, \text{kg m/s} \]

### Step 3: Write down the final momentum
After the collision, the final velocity of cart 1 (v1) is given as -1.0 m/s (moving to the left). Let the final velocity of cart 2 (v2) be V (which we need to find). The total final momentum (p_final) is:
\[ p_{\text{final}} = m_1 \cdot v_1 + m_2 \cdot v_2 \]
Substituting the known values:
\[ p_{\text{final}} = (3.0 \, \text{kg} \cdot -1.0 \, \text{m/s}) + (5.0 \, \text{kg} \cdot V) \]
\[ p_{\text{final}} = -3.0 \, \text{kg m/s} + 5.0 \, \text{kg} \cdot V \]

### Step 4: Set initial momentum equal to final momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Thus:
\[ -7.0 \, \text{kg m/s} = -3.0 \, \text{kg m/s} + 5.0 \, \text{kg} \cdot V \]

### Step 5: Solve for V
Rearranging the equation:
\[ -7.0 + 3.0 = 5.0 \cdot V \]
\[ -4.0 = 5.0 \cdot V \]
\[ V = \frac{-4.0}{5.0} = -0.8 \, \text{m/s} \]

### Conclusion
The final velocity of the 5.0 kg cart is -0.8 m/s, indicating that it is moving to the left.