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?

Verified step by step guidance

1. Calculate the time it would take for the asteroid to reach Earth if the rocket is not fired. Use the formula for time, which is the distance divided by the velocity. Here, the distance is 4.0 x 10^6 km and the velocity is 20 km/s.
2. Convert the distance from kilometers to meters by multiplying by 1000, as the velocity is given in meters per second.
3. Convert the time from seconds to hours by dividing by 3600, since there are 3600 seconds in an hour.
4. Check the units of your calculation to ensure they are consistent, especially ensuring that distance and velocity units match.
5. Summarize the steps and ensure all calculations are correct to find the time in hours until the asteroid impacts the Earth if the rocket does not alter its course.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and displacement. In this scenario, understanding the initial velocity of the asteroid and the distance it needs to travel to reach Earth is crucial for calculating the time until impact.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as F = ma. This principle is essential for analyzing the effect of the rocket's thrust on the asteroid's motion. Although the question asks to ignore gravitational forces, the thrust will alter the asteroid's trajectory and speed, which is vital for understanding the overall dynamics.

Time Calculation

Time calculation in physics often involves determining how long it takes for an object to travel a certain distance at a given speed. In this case, the time until the asteroid impacts Earth can be calculated using the formula time = distance / speed. By applying this formula to the initial conditions of the asteroid's motion, one can find the time until impact, assuming no other forces act on it.