Question

A NASA explorer spacecraft with a mass of 1,000 kg takes off in a positive direction from a stationary asteroid.

If the velocity of the spacecraft is 250 m/s and the asteroid is pushed back –25 m/s, what is the mass of the asteroid? Assume there is no net force on the system.

100 kg
6,250 kg
10,000 kg
6,250,000 kg

Answer 1

To solve the problem, we can use the law of conservation of momentum, which states that in the absence of external forces, the total momentum of a closed system remains constant.

Let's denote:
- \( m_s \) = mass of the spacecraft = 1,000 kg
- \( v_s \) = velocity of the spacecraft = 250 m/s
- \( m_a \) = mass of the asteroid (unknown)
- \( v_a \) = velocity of the asteroid = -25 m/s (negative since it’s moving in the opposite direction)

According to the conservation of momentum:

\[
m_s \cdot v_s + m_a \cdot v_a = 0
\]

Substituting the known values into the equation:

\[
1000 \, \text{kg} \cdot 250 \, \text{m/s} + m_a \cdot (-25 \, \text{m/s}) = 0
\]

Calculating the momentum of the spacecraft:

\[
1000 \cdot 250 = 250000 \, \text{kg m/s}
\]

So the equation becomes:

\[
250000 - 25m_a = 0
\]

Rearranging the equation to solve for \( m_a \):

\[
25m_a = 250000
\]

\[
m_a = \frac{250000}{25} = 10000 \, \text{kg}
\]

The mass of the asteroid is **10,000 kg**.