Question

A 4.0 × 10¹⁰ kg asteroid is heading directly toward the center of the earth at a steady 20 km/s. To save the planet, astronauts strap a giant rocket to the asteroid perpendicular to its direction of travel. The rocket generates 5.0 × 10⁹ N of thrust. The rocket is fired when the asteroid is 4.0 x 10⁶ km away from the earth. You can ignore the earth's gravitational force on the asteroid and their rotation around the sun. If the mission fails, how many hours is it until the asteroid impacts the earth?

Answer 1

Calculating the time for an asteroid traveling at 20 km/s to cover a distance of 4.0 x 10⁶ km, we find the impact would occur after approximately 5555.56 hours if a mission to change its course fails.

Step-by-step explanation:

The question is a physics problem involving the motion of an asteroid toward Earth and considers what would happen if a mission to alter the asteroid's trajectory fails.

To find out how many hours until the asteroid impacts Earth, we need to calculate the time taken for an asteroid traveling at a constant speed to cover a given distance. The asteroid is moving at 20 km/s, and it is 4.0 x 10⁶ km away from Earth. We can use the formula Δt = d/v, where Δt is the time, d is the distance, and v is the velocity.

The calculation is as follows:

Convert the distance from kilometers to meters: 4.0 x 10⁶ km * 10³ m/km = 4.0 x 10⁹ m.

Calculate the time in seconds: Δt = 4.0 x 10⁹ m / (20 x 10³ m/s) = 2.0 x 10⁵ s.

Convert the time from seconds to hours: 2.0 x 10⁵ s / (3600 s/hr) ≈ 5555.56 hours.

Therefore, if the mission fails, the asteroid would impact Earth after approximately 5555.56 hours.