The bullet in the previous problem strikes a $$ 2.5-\mathrm{kg} $$ steel ball that is at rest. The bullet bounces backward after its collision at a speed of $$ 5.0 \mathrm{~m} / \mathrm{s} $$. How fast is the ball moving when the bullet bounces backward?

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To solve this problem, we can use the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision.

### Known Conditions:
- Mass of the steel ball, $$ m_b = 2.5 \, \text{kg} $$
- Initial velocity of the steel ball, $$ v_{b_i} = 0 \, \text{m/s} $$ (at rest)
- Mass of the bullet, $$ m_{bu} $$ (unknown)
- Initial velocity of the bullet, $$ v_{bu_i} $$ (unknown)
- Final velocity of the bullet after the collision, $$ v_{bu_f} = -5.0 \, \text{m/s} $$ (negative because it bounces backward)
- Final velocity of the steel ball, $$ v_{b_f} $$ (unknown)

### Step-by-Step Solution:

1. **Write the conservation of momentum equation:**
   $$
   m_{bu} v_{bu_i} + m_b v_{b_i} = m_{bu} v_{bu_f} + m_b v_{b_f}
   $$

Since the steel ball is at rest initially, $$ v_{b_i} = 0 $$:
   $$
   m_{bu} v_{bu_i} = m_{bu} (-5.0) + 2.5 v_{b_f}
   $$

2. **Rearranging the equation:**
   $$
   m_{bu} v_{bu_i} + 5.0 m_{bu} = 2.5 v_{b_f}
   $$
   $$
   m_{bu} (v_{bu_i} + 5.0) = 2.5 v_{b_f}
   $$

3. **To find $$ v_{b_f} $$, we need the mass of the bullet $$ m_{bu} $$ and its initial velocity $$ v_{bu_i} $$.** However, since we don't have these values, we can express $$ v_{b_f} $$ in terms of $$ m_{bu} $$ and $$ v_{bu_i} $$:
   $$
   v_{b_f} = \frac{m_{bu} (v_{bu_i} + 5.0)}{2.5}
   $$

4. **If we assume a specific value for the bullet's mass and initial velocity, we can calculate $$ v_{b_f} $$.** For example, if we assume $$ m_{bu} = 0.01 \, \text{kg} $$ (10 grams) and $$ v_{bu_i} = 20 \, \text{m/s} $$:
   $$
   v_{b_f} = \frac{0.01 (20 + 5)}{2.5}
   $$

Let's calculate $$ v_{b_f} $$ using these assumed values.

### Calculation:
$$
v_{b_f} = \frac{0.01 (25)}{2.5} = \frac{0.25}{2.5} = 0.1 \, \text{m/s}
$$

Thus, the steel ball would be moving at $$ 0.1 \, \text{m/s} $$ when the bullet bounces backward, given the assumed values for the bullet's mass and initial velocity.

If you have specific values for the bullet's mass and initial velocity, please provide them for a more accurate calculation.
The steel ball moves at 0.1 m/s when the bullet bounces backward.

The bullet in the previous problem strikes a \( 2.5-\mathrm{kg} \) steel ball that is at rest. The bullet bounces backward after its collision at a speed of \( 5.0 \mathrm{~m} / \mathrm{s} \). How fast is the ball moving when the bullet bounces backward?

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