Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 47a

Chapter 6, Problem 47a

A rocket of mass m is launched straight up with thrust F_thrust. Find an expression for the rocket's speed at height h if air resistance is neglected.

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Start by identifying the forces acting on the rocket. The upward force is the thrust \( F_{\text{thrust}} \), and the downward force is the gravitational force \( F_{\text{gravity}} = m g \), where \( m \) is the mass of the rocket and \( g \) is the acceleration due to gravity.

Write the net force acting on the rocket using Newton's second law: \( F_{\text{net}} = F_{\text{thrust}} - F_{\text{gravity}} = m a \), where \( a \) is the rocket's acceleration.

Rearrange the equation to solve for the acceleration: \( a = \frac{F_{\text{thrust}} - m g}{m} \). This gives the rocket's acceleration as a function of the thrust and gravitational force.

Use the kinematic equation \( v^2 = v_0^2 + 2 a h \) to find the rocket's speed at height \( h \). Assuming the rocket starts from rest (\( v_0 = 0 \)), the equation simplifies to \( v^2 = 2 a h \).

Substitute the expression for \( a \) into the kinematic equation: \( v^2 = 2 \left( \frac{F_{\text{thrust}} - m g}{m} \right) h \). Finally, take the square root to find the speed: \( v = \sqrt{\frac{2 h (F_{\text{thrust}} - m g)}{m}} \).

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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 analyzing the motion of the rocket, as the thrust force must overcome both the gravitational force and any inertial effects to determine the rocket's acceleration.

Kinematics of Motion

Kinematics involves the study of motion without considering the forces that cause it. In this context, we can use kinematic equations to relate the rocket's initial velocity, acceleration, and displacement (height h) to find its final speed. Understanding these relationships is essential for deriving the rocket's speed at a given height.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the case of the rocket, the work done by the thrust force converts into kinetic energy and potential energy as the rocket ascends. This concept helps in deriving the relationship between the rocket's speed and height by equating the work done to the change in energy.

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The motor of a 350 g model rocket generates 9.5 N thrust. If air resistance can be neglected, what will be the rocket's speed as it reaches a height of 85 m?

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Textbook Question

A rocket of mass m is launched straight up with thrust Fthrust. Find an expression for the rocket's speed at height h if air resistance is neglected.

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Key Concepts

Newton's Second Law of Motion

Kinematics of Motion

Conservation of Energy

A rocket of mass m is launched straight up with thrust F_thrust. Find an expression for the rocket's speed at height h if air resistance is neglected.