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Ch 08: Dynamics II: Motion in a Plane
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 08: Dynamics II: Motion in a PlaneProblem 4b
Chapter 8, Problem 4b

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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Step 1: Identify the key quantities given in the problem. The asteroid has a mass of 4.0 × 10^10 kg, an initial velocity of 20 km/s, and is initially 4.0 × 10^6 km away from Earth. The rocket generates a thrust of 5.0 × 10^9 N perpendicular to the asteroid's direction of travel. The radius of the Earth is 6400 km, and we need to calculate the minimum angle of deflection for the asteroid to just miss the Earth.
Step 2: Calculate the acceleration produced by the rocket's thrust. Use Newton's second law, F = ma, where F is the thrust and m is the mass of the asteroid. The acceleration a can be expressed as: a=Fm. Substitute F = 5.0 × 10^9 N and m = 4.0 × 10^10 kg to find the acceleration.
Step 3: Determine the time it takes for the asteroid to reach Earth. Use the formula for time, t=dv, where d is the initial distance to Earth (4.0 × 10^6 km) and v is the asteroid's velocity (20 km/s). Convert the distance to meters and calculate the time.
Step 4: Calculate the deflection distance caused by the rocket's thrust. The deflection distance is determined by the lateral displacement due to the perpendicular acceleration. Use the kinematic equation for displacement: x=12at2, where a is the acceleration calculated in Step 2 and t is the time calculated in Step 3.
Step 5: Calculate the minimum angle of deflection. The angle can be determined using trigonometry. The tangent of the angle is given by tan(θ)=xd, where x is the deflection distance and d is the initial distance to Earth. Solve for θ using the inverse tangent function: θ=tan⁻(xd). Ensure the deflection angle is sufficient for the asteroid to miss the Earth's radius of 6400 km.

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Key Concepts

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

Momentum and Impulse

Momentum is the product of an object's mass and its velocity, representing its motion. Impulse is the change in momentum resulting from a force applied over 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 collision with Earth.
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Thrust and Acceleration

Thrust is the force exerted by a rocket engine to propel an object, calculated using Newton's second law (F = ma). The thrust from the rocket will cause the asteroid to accelerate perpendicular to its initial path. Understanding how thrust translates into acceleration is crucial for determining how much the asteroid's path will change and the angle required to miss Earth.
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Trajectory and Angle of Deflection

The trajectory of an object is the path it follows through space, influenced by its initial velocity and any forces acting on it. The angle of deflection is the angle at which the asteroid's path must be altered to avoid collision with Earth. By calculating the necessary change in trajectory due to the rocket's thrust, we can determine the minimum angle required for the asteroid to just miss the planet.
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A 1500 kg car takes a 50-m-radius unbanked curve at 15 m/s. What is the size of the friction force on the car?

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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. If the mission fails, how many hours is it until the asteroid impacts the earth?

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Textbook Question

As a science fair project, you want to launch an 800 g model rocket straight up and hit a horizontally moving target as it passes 30 m above the launch point. The rocket engine provides a constant thrust of 15.0 N. The target is approaching at a speed of 15 m/s. At what horizontal distance between the target and the rocket should you launch?

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Textbook Question

A 500 g model rocket is on a cart that is rolling to the right at a speed of. The rocket engine, when it is fired, exerts an 8.0 N vertical thrust on the rocket. Your goal is to have the rocket pass through a small horizontal hoop that is 20 m above the ground. At what horizontal distance left of the hoop should you launch?

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Textbook Question

A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. What is the speed of the block?

Textbook Question

In the Bohr model of the hydrogen atom, an electron (mass m = 9.1 x 10-31 kg) orbits a proton at a distance of 5.3 x 10-11 m. The proton pulls on the electron with an electric force of 8.2 x 10-8 N. How many revolutions per second does the electron make?