4. A time traveling brontosaurus with a mass of $$ 22,226 \mathrm{~kg} $$ is running with a velocity of $$ 75 \mathrm{~m} / \mathrm{s} $$ East. It collides head-on with a semi-truck with a mass of $$ 9,100 \mathrm{~kg} $$ moving in the opposite direction of $$ 120 \mathrm{~m} / \mathrm{s} $$ (west), which results in an elastic collision, sending the semi-truck backward with a final velocity of $$ 5 \mathrm{~m} / $$ s(east). What is the final velocity of the brontosaurus?

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To solve this problem, we will use the principles of conservation of momentum and the fact that the collision is elastic. In an elastic collision, both momentum and kinetic energy are conserved.

### Step 1: Define the known quantities

- Mass of the brontosaurus, $$ m_b = 22,226 \, \text{kg} $$
- Initial velocity of the brontosaurus, $$ v_{b_i} = 75 \, \text{m/s} $$ (East)
- Mass of the semi-truck, $$ m_t = 9,100 \, \text{kg} $$
- Initial velocity of the semi-truck, $$ v_{t_i} = -120 \, \text{m/s} $$ (West, hence negative)
- Final velocity of the semi-truck, $$ v_{t_f} = 5 \, \text{m/s} $$ (East)

### Step 2: Apply the conservation of momentum

The total momentum before the collision must equal the total momentum after the collision. The equation for momentum conservation is:

$$
m_b v_{b_i} + m_t v_{t_i} = m_b v_{b_f} + m_t v_{t_f}
$$

Where:
- $$ v_{b_f} $$ is the final velocity of the brontosaurus.

### Step 3: Substitute the known values into the equation

Substituting the known values into the momentum conservation equation:

$$
22,226 \cdot 75 + 9,100 \cdot (-120) = 22,226 \cdot v_{b_f} + 9,100 \cdot 5
$$

### Step 4: Calculate the left side of the equation

Calculating the left side:

1. Calculate $$ 22,226 \cdot 75 $$
2. Calculate $$ 9,100 \cdot (-120) $$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$22226\times 75$$
- step1: Multiply the numbers:
  $$1666950$$
Calculate or simplify the expression $$ 9,100 * (-120) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$9100\left(-120\right)$$
- step1: Use the rules for multiplication and division:
  $$-9100\times 120$$
- step2: Multiply the numbers:
  $$-1092000$$
The calculations yield the following results:

1. $$ 22,226 \cdot 75 = 1,666,950 $$
2. $$ 9,100 \cdot (-120) = -1,092,000 $$

### Step 5: Substitute the results back into the momentum equation

Now substituting these results into the momentum conservation equation:

$$
1,666,950 - 1,092,000 = 22,226 \cdot v_{b_f} + 9,100 \cdot 5
$$

### Step 6: Calculate the right side of the equation

First, calculate $$ 9,100 \cdot 5 $$:

$$
9,100 \cdot 5 = 45,500
$$

Now, substituting this into the equation gives:

$$
1,666,950 - 1,092,000 = 22,226 \cdot v_{b_f} + 45,500
$$

### Step 7: Simplify the left side

Calculating the left side:

$$
1,666,950 - 1,092,000 = 574,950
$$

### Step 8: Set up the equation to solve for $$ v_{b_f} $$

Now we have:

$$
574,950 = 22,226 \cdot v_{b_f} + 45,500
$$

### Step 9: Isolate $$ v_{b_f} $$

Subtract $$ 45,500 $$ from both sides:

$$
574,950 - 45,500 = 22,226 \cdot v_{b_f}
$$

Calculating the left side:

$$
574,950 - 45,500 = 529,450
$$

Now we have:

$$
529,450 = 22,226 \cdot v_{b_f}
$$

### Step 10: Solve for $$ v_{b_f} $$

Now, divide both sides by $$ 22,226 $$:

$$
v_{b_f} = \frac{529,450}{22,226}
$$

Let's calculate this final value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{529450}{22226}$$
- step1: Reduce the fraction:
  $$\frac{264725}{11113}$$
The final calculation gives us:

$$
v_{b_f} = \frac{529,450}{22,226} \approx 23.82 \, \text{m/s}
$$

This can also be expressed as:

$$
v_{b_f} = 23 \frac{9126}{11113} \, \text{m/s}
$$

### Conclusion

The final velocity of the brontosaurus after the elastic collision is approximately $$ 23.82 \, \text{m/s} $$ (East).
The final velocity of the brontosaurus is approximately $$ 23.82 \, \text{m/s} $$ East.

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