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Let’s analyze and solve the questions presented:

### Question A: Initial Momentum of the Carts Before Collision
1. **Given dataLet’s analyze and solve the questions presented:

### Question A: Initial Momentum of the Carts Before Collision
1. **Given **
   - Mass of Cart A = Mass of Cart B = 98 kg
   - Velocity of Cart A (\( v_A \)) = 3.0 m/s (moving to the right)
   - Velocity of Cart B (\( v_B \)) = -1.5 m/s (moving to the left)

2. **Calculate Initial Momentum of Each Cart:**
   - Momentum of Cart A (\( p_A \)) = \( m \times v_A = 98 \times 3.0 = 294 \, \text{kg m/s} \)
   - Momentum of Cart B (\( p_B \)) = \( m \times v_B = 98 \times (-1.5) = -147 \, \text{kg m/s} \)

3. **Total Initial Momentum:**
   - Total momentum = \( p_A + p_B = 294 + (-147) = 147 \, \text{kg m/s} \)

So, the correct answer for the initial momentum is **147 kg m/s**.

### Question B: Maximum Velocity of Cart A After Collision
Given that Cart B moves backward at 0.75 m/s after the collision, we want to find the maximum velocity of Cart A.

1. **Using Conservation of Momentum:**
   - The total momentum before the collision is 147 kg m/s, and this must be conserved.
   - Let the velocity of Cart A after the collision be \( v_{A}' \).
   - Momentum of Cart B after collision (\( p_B' \)) = \( m \times v_{B}' = 98 \times (-0.75) = -73.5 \, \text{kg m/s} \).

2. **Calculate \( v_{A}' \):**
   - Total momentum after collision = Momentum of Cart A + Momentum of Cart B
   - \( 147 = 98 \times v_{A}' + (-73.5) \)
   - Solving for \( v_{A}' \): 
     \[
     147 + 73.5 = 98 \times v_{A}' \Rightarrow v_{A}' = \frac{220.5}{98} = 2.25 \, \text{m/s}
     \]

Therefore, the maximum velocity of Cart A after the collision is **2.25 m/s**.

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Would you like further details on these calculations or have any other questions?

### Further Questions to Consider
1. How would the initial momentum change if Cart B had a different mass?
2. What would happen if this collision were inelastic rather than elastic?
3. How does conservation of momentum apply in different types of collisions?
4. How would you calculate the kinetic energy of each cart before and after the collision?
5. Can we determine the energy lost or gained in this collision?

### Tip
In elastic collisions, both momentum and kinetic energy are conserved. This can simplify solving for final velocities.

A 98 kg cart moving at 3.0 m/s has a head-on elastic collision with an equal mass cart moving at 1.5 m/s. 
A. What is the initial momentum of the carts before the collision? 
B. If cart B moves backwards at a speed of 0.75 m/s, what is the maximum velocity of cart A after the collision?

Learn how to calculate the initial momentum and final velocities of two carts in a head-on elastic collision, where each cart has a mass of 98 kg, and one cart moves at 3.0 m/s while the other moves at 1.5 m/s. Understand the conservation of momentum and apply it to find the maximum velocity of Cart A after the collision.

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