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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Convert the mass of the rocket from grams to kilograms: \( m = 800 \, \text{g} = 0.8 \, \text{kg} \).

Calculate the net force acting on the rocket using \( F_{\text{net}} = F_{\text{thrust}} - F_{\text{gravity}} \), where \( F_{\text{gravity}} = m \cdot g \) and \( g = 9.8 \, \text{m/s}^2 \).

Determine the rocket's acceleration using Newton's second law: \( a = \frac{F_{\text{net}}}{m} \).

Use the kinematic equation \( h = \frac{1}{2} a t^2 \) to solve for the time \( t \) it takes for the rocket to reach the height of 30 m. Rearrange the equation to \( t = \sqrt{\frac{2h}{a}} \).

Calculate the horizontal distance the target travels during the time \( t \) using \( d = v \cdot t \), where \( v = 15 \, \text{m/s} \). This distance is the horizontal separation at which the rocket should be launched.

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Key Concepts

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

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. This principle is crucial for understanding how the rocket will accelerate upwards under the influence of the thrust provided by its engine, allowing us to calculate its motion over time.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational force. In this scenario, the rocket's vertical motion can be analyzed as a projectile, allowing us to determine how long it will take to reach the target's height of 30 m, which is essential for calculating the horizontal distance needed for the launch.

Relative Velocity

Relative velocity is the velocity of an object as observed from another moving object. In this problem, understanding the relative velocity between the rocket and the target is key to determining the correct launch distance, as the target is moving horizontally while the rocket ascends vertically, necessitating a calculation of their positions over time.

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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

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 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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