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Ch. 09 - Linear Momentum
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 09 - Linear MomentumProblema 73
Capitolo 9, Problema 73

A rocket traveling 1950 m/s away from the Earth at an altitude of 6400 km fires its rockets, which eject gas at a speed of 1200 m/s (relative to the rocket). If the mass of the rocket at this moment is 25,000 kg and an acceleration of 1.5 m/s² is desired, at what rate must the gases be ejected?

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Step 1: Identify the key variables in the problem. The rocket's mass is \( m = 25,000 \, \text{kg} \), the desired acceleration is \( a = 1.5 \, \text{m/s}^2 \), and the relative velocity of the ejected gas is \( v_{\text{rel}} = 1200 \, \text{m/s} \). The goal is to find the rate of gas ejection \( \dot{m} \) (mass flow rate).
Step 2: Recall the thrust equation from Newton's second law for a rocket: \( F = \dot{m} v_{\text{rel}} \), where \( F \) is the force required to achieve the desired acceleration. The force can also be expressed as \( F = m a \). Equating these two expressions gives \( \dot{m} v_{\text{rel}} = m a \).
Step 3: Rearrange the equation to solve for \( \dot{m} \), the mass flow rate: \( \dot{m} = \frac{m a}{v_{\text{rel}}} \). This equation relates the mass flow rate to the rocket's mass, desired acceleration, and the relative velocity of the ejected gas.
Step 4: Substitute the given values into the equation: \( \dot{m} = \frac{25,000 \times 1.5}{1200} \). Perform the division to calculate the mass flow rate.
Step 5: Interpret the result. The calculated \( \dot{m} \) represents the rate at which the rocket must eject gas (in \( \text{kg/s} \)) to achieve the desired acceleration of \( 1.5 \, \text{m/s}^2 \).

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Rocket Propulsion

Rocket propulsion is the process by which a rocket moves by expelling mass in the form of gas. According to Newton's third law, for every action, there is an equal and opposite reaction; thus, the ejection of gas generates thrust that propels the rocket forward. The effectiveness of this propulsion depends on the speed of the ejected gas and the mass of the rocket.
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Thrust and Acceleration

Thrust is the force exerted by the rocket engines to propel the rocket upward or forward. It can be calculated using Newton's second law, where thrust equals mass times acceleration (F = ma). To achieve a desired acceleration, the thrust must be sufficient to overcome both the gravitational force and any additional forces acting on the rocket.
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Mass Flow Rate

Mass flow rate is the amount of mass ejected per unit time, crucial for determining how much thrust a rocket can generate. It is calculated by dividing the thrust by the effective exhaust velocity of the gases. In this scenario, the mass flow rate must be adjusted to achieve the desired acceleration while considering the speed at which the gases are expelled relative to the rocket.
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