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 47b

Chapter 6, Problem 47b

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?

Verified step by step guidance

Step 1: Identify the given values and relevant equations. The mass of the rocket is 350 g (convert to kilograms: 0.350 kg), the thrust force is 9.5 N, and the height reached is 85 m. Use the work-energy principle to solve the problem, where the work done by the thrust force is converted into the rocket's kinetic energy.

Step 2: Calculate the net force acting on the rocket. Since air resistance is neglected, the net force is the thrust minus the gravitational force. Gravitational force is given by \( F_g = m \cdot g \), where \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)). Subtract \( F_g \) from the thrust to find the net force.

Step 3: Determine the work done by the net force. Work is calculated using \( W = F_{\text{net}} \cdot d \), where \( F_{\text{net}} \) is the net force and \( d \) is the distance (height) the rocket travels, which is 85 m.

Step 4: Relate the work done to the kinetic energy of the rocket. The work-energy theorem states that \( W = \Delta KE \), where \( \Delta KE \) is the change in kinetic energy. Since the rocket starts from rest, its initial kinetic energy is zero, and \( \Delta KE = \frac{1}{2} m v^2 \). Solve for \( v \) (the rocket's speed) using \( v = \sqrt{\frac{2W}{m}} \).

Step 5: Substitute the values for \( W \) and \( m \) into the equation \( v = \sqrt{\frac{2W}{m}} \) to find the rocket's speed. Ensure all units are consistent (e.g., mass in kilograms, distance in meters, force in newtons) before performing the calculation.

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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 can be expressed with the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. In the context of the rocket, the thrust generated by the motor provides the net force that will accelerate the rocket upwards.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For the rocket's ascent, we can use these equations to determine the final speed at a specific height, taking into account the initial conditions and the acceleration due to the thrust.

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 converts chemical energy from the fuel into kinetic energy and gravitational potential energy as the rocket ascends. At the maximum height, the kinetic energy will be converted into potential energy, allowing us to analyze the rocket's speed at that height.

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Textbook Question

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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