A 4.0 x 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 x 109 N of thrust. The rocket is fired when the asteroid is 4.0 x 106 km away from earth. You can ignore the earth's gravitational force on the asteroid and their rotation about the sun. If the mission fails, how many hours is it until the asteroid impacts the earth?

Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 4a

Chapter 8, Problem 4a

A 4.0 x 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 x 10⁹ N of thrust. The rocket is fired when the asteroid is 4.0 x 10⁶ km away from earth. You can ignore the earth's gravitational force on the asteroid and their rotation about the sun. If the mission fails, how many hours is it until the asteroid impacts the earth?

Verified step by step guidance

Step 1: Calculate the time it would take for the asteroid to reach Earth if it continues traveling at its current velocity. Use the formula for time, \( t = \frac{d}{v} \), where \( d \) is the distance to Earth (\( 4.0 \times 10^6 \; \text{km} \)) and \( v \) is the velocity of the asteroid (\( 20 \; \text{km/s} \)). Convert the time from seconds to hours by dividing by 3600.

Step 2: Convert the given distance \( d \) from kilometers to meters to ensure consistent units. Since \( 1 \; \text{km} = 1000 \; \text{m} \), multiply \( 4.0 \times 10^6 \; \text{km} \) by 1000 to get the distance in meters.

Step 3: Substitute the converted distance (in meters) and the velocity (\( 20 \; \text{km/s} \), converted to \( 20000 \; \text{m/s} \)) into the formula \( t = \frac{d}{v} \) to calculate the time in seconds.

Step 4: Convert the time from seconds to hours by dividing the result by 3600 (since there are 3600 seconds in an hour). This will give the time until the asteroid impacts the Earth in hours.

Step 5: Interpret the result to understand how much time remains before the asteroid impacts Earth if no intervention occurs. This value represents the time available for any corrective actions to be taken.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In this scenario, understanding kinematics is essential to calculate the time it takes for the asteroid to travel a certain distance at a constant speed. The basic formula used is time = distance / speed, which allows us to determine how long it will take for the asteroid to reach Earth if no other forces act on it.

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. Although the question specifies ignoring gravitational forces, this law is crucial for understanding how the thrust from the rocket will affect the asteroid's motion if the mission were to succeed. It helps in analyzing the changes in velocity and direction due to the applied force.

Thrust and Impulse

Thrust is the force exerted by a rocket engine to propel an object, while impulse is the change in momentum resulting from a force applied over time. In this scenario, the thrust generated by the rocket can alter the asteroid's trajectory, but if the mission fails, the initial conditions of motion remain unchanged. Understanding thrust and impulse is vital for evaluating the potential effectiveness of the astronauts' intervention and the resulting motion of the asteroid.

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Textbook Question

A 4.0 x 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 x 10⁹ N of thrust. The rocket is fired when the asteroid is 4.0 x 10⁶ km away from earth. You can ignore the earth's gravitational force on the asteroid and their rotation about the sun.The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth?

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