Problem 46 - Physics for Scientists and Engineers with Modern Physics

Question

(1) Jill and her friends have built a small rocket that soon after lift-off reaches a speed of 850 \(\mathrm{m} / \mathrm{s} .\) How high above the Earth can it rise? Ignore air friction.

Step-by-Step Solution

Answer

The rocket can rise approximately 36862 meters.

Step 1: Understand the Problem

We want to find out how high a rocket can rise given its initial speed. The rocket has been launched vertically, and we are ignoring all forms of air resistance.

Step 2: Identify Relevant Physics Concepts

The problem involves gravitational potential energy, kinetic energy, and the work-energy principle. As the rocket rises, its kinetic energy is converted into gravitational potential energy until the speed becomes zero at the maximum height.

Step 3: Apply the Energy Conservation Principle

According to the law of conservation of energy, the initial kinetic energy of the rocket will be equal to the gravitational potential energy at its maximum height. The equations are given by:Initial Kinetic Energy: \( KE_i = \frac{1}{2} m v^2 \) Potential Energy at Maximum Height: \( PE_h = mgh \)Where:- \(m\) is the mass of the rocket (which will cancel out in the next step),- \(v\) is the velocity of the rocket = 850 m/s,- \(g\) is the acceleration due to gravity = 9.8 m/s²,- \(h\) is the height we need to find.

Step 4: Set Up the Equation

Since \(KE_i = PE_h\), we have:\[ \frac{1}{2} m v^2 = mgh \]Notice that the mass \(m\) cancels out, leaving us with:\[ \frac{1}{2} v^2 = gh \]

Step 5: Solve for Height, h

Rearrange the equation to find \(h\) as follows:\[ h = \frac{v^2}{2g} \]Substitute in the values:\[ h = \frac{(850)^2}{2 imes 9.8} \]Calculate it to find:\[ h \approx \frac{722500}{19.6} \approx 36862.24 \text{ meters} \]

Step 6: Final Check and Conclusion

Double-check the calculations and confirm the result makes sense. Thus, the rocket can rise approximately 36862.24 meters above the Earth.

Key Concepts

Kinetic Energy

Kinetic energy plays a crucial role when analyzing motion, as it is the energy possessed by an object due to its motion. Any object in motion has kinetic energy, and this energy increases with both mass and speed. The formula for calculating kinetic energy \[ KE = \frac{1}{2} m v^2 \] helps us determine the energy associated with moving objects, where:

• \( m \) is the mass of the object,
• \( v \) is its velocity.

In our problem, Jill's rocket has a velocity of 850 m/s at lift-off, which means it starts with a significant amount of kinetic energy.

Understanding kinetic energy means knowing that it is directly tied to speed: the faster something moves, the more kinetic energy it has. If you've ever wondered why it's harder to stop a fast-moving car than a slow bike, kinetic energy is the answer.
By comprehensively understanding kinetic energy, you gain insights into how energy is transferred between states — from motion to height or thermal energy, highlighting its foundational role in conservation of energy.

Gravitational Potential Energy

Gravitational potential energy is the energy stored due to an object's position in a gravitational field, typically due to its height above the ground. As an object ascends, it gains potential energy while losing kinetic energy. The relationship is mathematically described as:\[ PE = mgh \] Here,

• \( m \) is the mass,
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth),
• \( h \) is the height above the ground.

For the rocket, as it ascends, its speed decreases, converting kinetic energy into gravitational potential energy until it reaches its maximum height where speed equals zero.

Think of potential energy as stored energy that has the potential to do work — for example, water held in a dam at height, later released to generate electricity. The idea of potential energy helps us understand how and why energy can transform forms without loss, preserving the overall energy balance.

Work-Energy Principle

The work-energy principle is a key concept that ties together the ideas of work done on an object and the change in its energy. It essentially states that the work done on an object is equal to the change in its kinetic energy. This principle can be summarized as:\[ \text{Work Done} = \Delta KE \] In simpler terms, when work is done on an object, energy is transferred, often changing the object's speed. In the context of Jill's rocket, the initial kinetic energy is converted entirely into gravitational potential energy as the rocket rises. This principle allows us to understand and solve questions involving energy conservation without needing to measure every force individually.

The work-energy principle not only simplifies calculations but also provides a powerful tool for understanding how energy can move between different forms and systems. It's a cornerstone for studying both the motion of objects and broader physics topics, as it emphasizes the constancy of energy within a closed system.