Question

An astronaut circling the earth at an altitude of 400 km is horrified to discover that a cloud of space debris is moving in the exact same orbit as his spacecraft, but in the opposite direction. The astronaut detects the debris when it is 29 km away. How much time does he have to fire his rockets and change orbits?

Answer 1

To determine the time the astronaut has to fire his rockets and change orbits, calculate the relative velocity between the spacecraft and the space debris. The relative velocity is the sum of the velocities of the spacecraft and the space debris. Divide the distance between them by the relative velocity to find the time the astronaut has.

Step-by-step explanation:

To determine how much time the astronaut has to fire his rockets and change orbits, we need to calculate the relative velocity between the astronaut's spacecraft and the space debris. The relative velocity is the sum of the velocities of the astronaut's spacecraft and the space debris. Since the space debris is moving in the opposite direction, its velocity is negative.

The relative velocity can be calculated as follows:

Relative velocity = Velocity of spacecraft + Velocity of space debris = (Velocity of spacecraft) - (Velocity of space debris)

The distance between the astronaut's spacecraft and the space debris is given as 29 km. To calculate the time the astronaut has to fire his rockets, we divide this distance by the relative velocity:

Time = Distance / Relative velocity

Substituting the given values:

Distance = 29 km = 29,000 m

Altitude of the spacecraft = 400 km = 400,000 m

Relative velocity = (Velocity of spacecraft) - (Velocity of space debris)

We can assume the spacecraft's velocity is the circular orbital velocity at an altitude of 400 km, which can be calculated using the formula:

Velocity of spacecraft = sqrt(G * Mass of Earth / (Radius of Earth + Altitude of spacecraft))

Plugging in the values:

Mass of Earth = 5.97 * 10^24 kg

Radius of Earth = 6,371 km = 6,371,000 m

Using these values, we can calculate the velocity of the spacecraft:

Velocity of spacecraft = sqrt((6.67 * 10^-11 Nm^2/kg^2) * (5.97 * 10^24 kg) / ((6,371,000 m + 400,000 m)))

Once we have the velocity of the spacecraft, we can calculate the velocity of the space debris as:

Velocity of space debris = - Velocity of spacecraft

Finally, substituting the values into the equation for time:

Time = 29,000 m / (Velocity of spacecraft - Velocity of space debris)

Calculating this value will give us the amount of time the astronaut has to fire his rockets and change orbits.