To determine the mass of the asteroid, let's apply the principle of conservation of momentum. According to this principle, in the absence of external forces, the total momentum of a system remains constant.
We are given the following information:
- Mass of the spacecraft ([tex]\( m_s \)[/tex]) = 1000 kg
- Velocity of the spacecraft ([tex]\( v_s \)[/tex]) = 250 m/s (positive direction)
- Velocity of the asteroid ([tex]\( v_a \)[/tex]) = -25 m/s (negative direction)
Initially, the spacecraft and the asteroid are stationary, so their initial momenta are zero. After the spacecraft takes off, the momentum of the system must still sum to zero because no external force is acting on it.
Let's denote the mass of the asteroid by [tex]\( m_a \)[/tex].
The conservation of momentum can be written as:
[tex]\[ m_s \cdot v_s + m_a \cdot v_a = 0 \][/tex]
Substitute the known values into the equation:
[tex]\[ 1000 \cdot 250 + m_a \cdot (-25) = 0 \][/tex]
Solve for [tex]\( m_a \)[/tex]:
[tex]\[ 250000 - 25m_a = 0 \][/tex]
[tex]\[ -25m_a = -250000 \][/tex]
[tex]\[ m_a = \frac{250000}{25} \][/tex]
[tex]\[ m_a = 10000 \, \text{kg} \][/tex]
So, the mass of the asteroid is [tex]\(\boxed{10000 \, \text{kg}}\)[/tex].
