To solve this problem, we will apply the principle of conservation of momentum, since the planet is initially at rest and there are no external forces acting on it.

Step-by-step solution:

1. **Given dataTo solve this problem, we will apply the principle of conservation of momentum, since the planet is initially at rest and there are no external forces acting on it.

Step-by-step solution:

1. **Given **

   • Total mass of the planet, $M = 3 \times 10^{24} \, \text{kg}$.
   • 40% of the mass flies away in one piece, which means mass of this piece is: $m_1 = 0.4M = 0.4 \times 3 \times 10^{24} = 1.2 \times 10^{24} \, \text{kg}.$
   • Speed of this piece, $v_1 = 4000 \, \text{km/s} = 4 \times 10^6 \, \text{m/s}$.
   • The remaining mass is 60% of the planet’s mass, i.e.: $m_2 = 0.6M = 0.6 \times 3 \times 10^{24} = 1.8 \times 10^{24} \, \text{kg}.$

2. Conservation of momentum: Since the planet was initially at rest, the total momentum before the explosion was zero. After the explosion, the momentum of the two pieces must also sum to zero. Therefore: $m_1 v_1 + m_2 v_2 = 0,$ where $v_2$ is the velocity of the second piece (the remaining mass).

   Rearranging for $v_2$: $v_2 = -\frac{m_1 v_1}{m_2}.$

3. Substitute the known values: $v_2 = -\frac{1.2 \times 10^{24} \times 4 \times 10^6}{1.8 \times 10^{24}}.$ Simplifying: $v_2 = -\frac{4.8 \times 10^{30}}{1.8 \times 10^{24}} = -2.67 \times 10^6 \, \text{m/s}.$

   So, the speed of the remaining 60% of the planet is $2.67 \times 10^6 \, \text{m/s}$, or 2,670 km/s in the opposite direction.

Final answer:

The remaining 60% of the planet moves at 2,670 km/s in the direction opposite to the first piece.

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Would you like further clarification on any part of this? Here are 5 related questions you might find interesting:

1. How does the law of conservation of momentum apply to collisions?
2. What are some examples of explosions in space where this principle is applied?
3. How does relativistic momentum differ from classical momentum?
4. What would happen if external forces were present in this explosion?
5. How would the speeds change if the mass proportions were different?

Tip: Always ensure the units are consistent (e.g., converting km to m) when applying physical formulas.