A 4.0 x 10^10 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 x 10^9 N of thrust. The rocket is fired when the asteroid is 4.0 x 10^6 km away from earth. You can ignore the earth's gravitational force on the asteroid and their rotation about the sun. (a) If the mission fails, how many hours is it until the asteroid impacts the earth?

Video transcript

Hi, everyone in this practice problem. We are asked to determine when the rocket will crash on earth where we'll have a rocket with a mass of 5.49 times 10 to the power of five kg going back to the earth's orbit. However, when the rocket was at three times 10 to the power of five kilometers away, one of its engine actually failed due to a malfunction and it started falling back at a rate of 15 kilometers per second. We were asked to ignore the earth's gravitational force on the rocket and determine when the rocket will actually crash on the earth. The options are a eight times 10 to the power of two seconds. B, one times 10 to the power of 11 seconds. C three times 10 to the power of seven seconds and D two times 10 to the power of three seconds. So first, I'm going to, to start us off with making a list of everything that is given that are important for us to actually solve this problem. First, you'll have the displacement D which is just three times 10 to the power of 4 km because that is the position where the earth is experiencing or the rocket is experiencing the malfunction. Next, we have the velocity, the velocity is going to equals to 15 km/s. And that is essentially the rate at which the rocket is falling back to the earth. So we're being asked to uh determine the time that it takes for the rocket to essentially finally reach or fall or crash into the earth. So we want to use the equation of time equals displacement divided by velocity to essentially determine this value. So the time will then equals to D over V which will equals to three times 10 to the power of four kilometers divided by 15 kilometers per second, which will then actually correspond to two times 10 to the power of three seconds. And that will essentially be the answer to our problem, which will correspond to option D. So option D with the time of two times 10 to the power of three seconds is going to be the answer to this particular practice problem. And if you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics and they'll be all for this one. Thank you.