An asteroid, headed directly toward Earth, has a speed of 12 km/s relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed `v_f` when it reaches Earth's surface.

Explanation

To find the speed of the asteroid when it reaches Earth's surface, we can use the principle of conservation of mechanical energy. The total mechanical energy (kinetic + potential) remains constant if we neglect air resistance and other non-conservative forces.

### Step-by-step Solution:

1. **Identify Initial Conditions:**
- Initial speed of the asteroid, \( v_i = 12 \, \text{km/s} = 12 \times 10^3 \, \text{m/s} \)
- Initial distance from the center of the Earth, \( r_i = 10 R_E \)
- Radius of the Earth, \( R_E = 6.37 \times 10^6 \, \text{m} \)
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