Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 40a

Chapter 8, Problem 40a

Early test flights for the space shuttle used a “glider” (mass of 980 kg including pilot). After a horizontal launch at 480 km/h at a height of 3200 m, the glider eventually landed at sea level with a speed of 210 km/h. What would its landing speed have been in the absence of air resistance?

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Convert all given speeds from km/h to m/s for consistency in SI units. Use the conversion factor: \(1 \ \text{km/h} = \frac{1000}{3600} \ \text{m/s}\). For example, \(v_\text{initial} = 480 \ \text{km/h} \times \frac{1000}{3600}\) and \(v_\text{final} = 210 \ \text{km/h} \times \frac{1000}{3600}\).

Apply the principle of conservation of mechanical energy, assuming no air resistance. The total mechanical energy at the initial point (launch) is the sum of kinetic energy \(KE\) and potential energy \(PE\): \(E_\text{initial} = \frac{1}{2} m v_\text{initial}^2 + m g h\), where \(m\) is the mass of the glider, \(g\) is the acceleration due to gravity (\(9.8 \ \text{m/s}^2\)), and \(h\) is the height (3200 m).

At the landing point (sea level), the potential energy is zero because \(h = 0\). The total mechanical energy is purely kinetic: \(E_\text{final} = \frac{1}{2} m v_\text{final}^2\).

Set the initial total energy equal to the final total energy to solve for the landing speed in the absence of air resistance: \(\frac{1}{2} m v_\text{initial}^2 + m g h = \frac{1}{2} m v_\text{final}^2\). Notice that the mass \(m\) cancels out from all terms.

Rearrange the equation to solve for \(v_\text{final}\): \(v_\text{final} = \sqrt{v_\text{initial}^2 + 2 g h}\). Substitute the known values for \(v_\text{initial}\), \(g\), and \(h\) to calculate the landing speed in the absence of air resistance.

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Key Concepts

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

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In the context of the glider, the potential energy at its initial height is converted into kinetic energy as it descends. This relationship allows us to calculate the theoretical landing speed by equating the initial potential energy to the kinetic energy at sea level, assuming no energy is lost to air resistance.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. For the glider, its kinetic energy at landing can be determined by its speed just before impact. Understanding kinetic energy is crucial for determining how speed changes as the glider descends and accelerates towards the ground.

Potential Energy

Potential energy is the stored energy of an object due to its position in a gravitational field, calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. In this scenario, the glider's initial potential energy at 3200 m is converted into kinetic energy as it descends, which is essential for calculating the speed it would have landed at without air resistance.

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