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.(b) The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth?

Verified step by step guidance

1. Calculate the time it takes for the asteroid to reach Earth without the influence of the rocket. Use the formula for time, t = d / v, where d is the distance to Earth and v is the velocity of the asteroid.
2. Determine the acceleration provided by the rocket using Newton's second law, F = ma, where F is the force exerted by the rocket and m is the mass of the asteroid.
3. Calculate the time duration for which the rocket needs to be fired to change the velocity of the asteroid perpendicular to its original direction. Use the formula a = Δv / t, where Δv is the change in velocity and t is the time calculated in step 1.
4. Find the perpendicular distance the asteroid would travel due to the rocket's thrust while it is heading towards Earth. Use the formula d = 0.5 * a * t^2, where a is the acceleration from step 2 and t is the time from step 1.
5. Calculate the minimum deflection angle required to just miss the Earth using the tangent function, θ = tan⁻¹(opposite/adjacent), where the opposite side is the perpendicular distance from step 4 and the adjacent side is the initial distance to Earth.

Key Concepts

Momentum and Impulse

Momentum is the product of an object's mass and its velocity, representing the quantity of motion it possesses. Impulse is the change in momentum resulting from a force applied over a period of time. In this scenario, the thrust from the rocket will create an impulse that alters the asteroid's trajectory, allowing us to calculate the necessary deflection angle to avoid a collision with Earth.

Thrust and Acceleration

Thrust is the force exerted by a rocket engine to propel an object, calculated as the mass flow rate of the propellant multiplied by the velocity of the exhaust. According to Newton's second law, the acceleration of the asteroid can be determined by dividing the thrust by its mass. This acceleration will influence how much the asteroid's path can be altered over the distance it travels before reaching Earth.

Trajectory and Angle of Deflection

The trajectory of an object is the path it follows through space, which can be influenced by forces acting on it. The angle of deflection refers to the angle at which the asteroid's path must be altered to avoid collision with Earth. By calculating the necessary change in velocity due to the thrust and the distance to Earth, we can determine the minimum angle required for the asteroid to just miss the planet.