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This problem involves a system where a football player throws a football while moving on smooth ice. To solve this, we apply the **principle of conservation of momentum**.

### Part A: Player's speed after throwing the ball at 15.5 m/s relative to the ground

#### Step-by-step solution:
1. **Initial conditions**: The total momentum of the system (player + football) before the throw is the momentum of the player since the ball is being carried by him.
   - Mass of the player = \( m_p = 67.0 \, \text{kg} \)
   - Mass of the football = \( m_b = 0.430 \, \text{kg} \)
   - Initial velocity of both player and football = \( v_i = 1.75 \, \text{m/s} \)

2. **Final conditions**:
   - After the throw, the football moves at \( v_b = 15.5 \, \text{m/s} \) relative to the ground.
   - The player's velocity will change to \( v_f \), which we need to find.

3. **Conservation of momentum**:
   Since no external forces act on the system, the total momentum before and after the throw must be the same.

   **Initial momentum**:
   \[
   p_i = (m_p + m_b) v_i
   \]

   **Final momentum**:
   \[
   p_f = m_p v_f + m_b v_b
   \]

   Using the conservation of momentum equation:
   \[
   (m_p + m_b) v_i = m_p v_f + m_b v_b
   \]

4. **Solve for \( v_f \)**:
   \[
   v_f = \frac{(m_p + m_b) v_i - m_b v_b}{m_p}
   \]

   Substitute the given values:
   \[
   v_f = \frac{(67.0 + 0.430) \cdot 1.75 - 0.430 \cdot 15.5}{67.0}
   \]
   \[
   v_f = \frac{67.43 \cdot 1.75 - 6.665}{67.0}
   \]
   \[
   v_f = \frac{118.0025 - 6.665}{67.0} = \frac{111.3375}{67.0} \approx 1.661 \, \text{m/s}
   \]

So, the player's speed after the throw is approximately **1.66 m/s**.

Let me know if you'd like further details or explanations. Here are a few additional questions related to this topic:

1. What is the direction of the player's velocity after the throw relative to the initial velocity?
2. How does the mass of the football influence the player's final velocity?
3. What if the football was thrown in the opposite direction of the player's motion? How would that affect the final velocity?
4. How does the conservation of momentum principle apply in collisions?
5. What external forces would violate the conservation of momentum in this scenario?

**Tip**: Always ensure the system is isolated when applying the conservation of momentum, meaning no external forces (like friction) are acting on it.

A 67.0 kg football player is gliding across very smooth ice at 1.75 m/s. He throws a 0.430 kg football straight forward. What is the player's speed afterward if the ball is thrown at 15.5 m/s relative to the ground?

Learn how to apply the conservation of momentum to calculate the final speed of a 67.0 kg football player after throwing a 0.430 kg ball on smooth ice. Understand the system's initial and final momentum to solve this step-by-step problem involving kinematics.

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