Problem 55 - College Physics

Question

A rocket is fired in deep space, where gravity is negligible. If the rocket has an initial mass of 6000 \(\mathrm{kg}\) and ejects gas at a relative velocity of magnitude 2000 \(\mathrm{m} / \mathrm{s}\) , how much gas must it eject in the first second to have an initial acceleration of 25.0 \(\mathrm{m} / \mathrm{s}^{2}\) .

Step-by-Step Solution

Answer

The rocket must eject 75 kg of gas in the first second.

Step 1: Understanding the Problem

We need to find how much mass the rocket must eject in the first second to achieve a specified acceleration of 25 \( \mathrm{m/s}^2 \). The rocket operates in deep space where gravity is negligible, so the thrust produced by ejecting gas provides the required acceleration.

Step 2: Identify Key Variables

The initial mass of the rocket is \( m_0 = 6000 \ \mathrm{kg} \). The exhaust velocity of the gas is \( v_e = 2000 \ \mathrm{m/s} \). The desired acceleration is \( a = 25 \ \mathrm{m/s}^2 \). We need to find the mass \( \Delta m \) of the gas ejected in the first second.

Step 3: Relate Thrust to Acceleration

The thrust \( F \) produced by the rocket can be related to acceleration using Newton's second law \( F = ma \). Since \( a \) is the desired acceleration and \( m_0 \) is the initial mass of the rocket, we have \( F = m_0 \cdot a \).

Step 4: Calculate Required Thrust

Using \( m_0 = 6000 \ \mathrm{kg} \) and \( a = 25 \ \mathrm{m/s}^2 \), the required thrust is: \[ F = m_0 \cdot a = 6000 \cdot 25 = 150000 \ \mathrm{N} \]

Step 5: Using the Rocket Thrust Formula

The rocket thrust is also given by \( F = \Delta m \cdot v_e \), where \( \Delta m \) is the mass of gas ejected per unit time (per second here) and \( v_e \) is the exhaust velocity. We need to solve for \( \Delta m \).

Step 6: Solve for Ejected Mass

Set the expressions for thrust equal and solve for \( \Delta m \): \[ F = \Delta m \cdot v_e = 150000 \] \[ 150000 = \Delta m \cdot 2000 \] \[ \Delta m = \frac{150000}{2000} = 75 \ \mathrm{kg} \]

Step 7: Conclusion

The rocket must eject \( 75 \ \mathrm{kg} \) of gas in the first second to achieve the desired acceleration of \( 25 \ \mathrm{m/s}^2 \).

Key Concepts

Newton's Second Law

Newton's Second Law is fundamental in understanding motion, especially in scenarios involving forces. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \( F = ma \). This equation highlights the relationship between force, mass, and acceleration. In the context of rocket propulsion, this law explains how changing the mass or acceleration will affect the force needed for motion. When a rocket in space needs to accelerate, the force that enables this motion is produced by the ejection of gas. The force acts on the rocket, propelling it forward. In our exercise, the rocket's initial mass and desired acceleration are known, allowing us to use Newton's Second Law to calculate the force required. Understanding this equation is crucial for designing rockets and predicting their behavior in space, where other forces like gravity are negligible.

Mass Ejection

Mass ejection is the process by which a rocket expels part of its mass in the form of gas to generate thrust. This ejection is crucial for propulsion, especially in the vacuum of space. The expelled gas creates an equal and opposite reaction on the rocket due to Newton's Third Law, which provides the thrust necessary to propel the rocket.In deep space, where there is no atmosphere to assist, the rocket must rely entirely on the mass ejection for movement. The amount of mass ejected (\( \Delta m \)) is directly linked to the rocket's ability to accelerate. In our scenario, the rocket must eject a specific amount of mass per second to achieve the targeted acceleration of \( 25 \, \mathrm{m/s}^2 \), calculated based on the known exhaust velocity and required force. Efficient management of this ejected mass is vital for conserving fuel and ensuring the rocket reaches its destination.

Acceleration Calculation

Calculating a rocket's acceleration involves understanding the forces at play and the resulting movement. The desired acceleration is part of the equation in designing the thrust requirements. Using Newton's Second Law, where force is the product of mass and acceleration, you can calculate the acceleration within the parameters set by the thrust and mass.For our rocket, with a known initial mass of \( 6000 \, \mathrm{kg} \) and a specified acceleration of \( 25 \, \mathrm{m/s}^2 \), we focus on determining how much mass needs to be ejected to achieve the needed force to meet that acceleration. By setting the required force as \( 150000 \, \mathrm{N} \), calculated from multiplying the mass and targeted acceleration, we established the necessary conditions to work out other unknowns in rocket equations. This approach ensures precise results, optimizing fuel use and adjusting to mission specifications.

Exhaust Velocity

Exhaust velocity refers to the speed at which gas is expelled from the rocket, playing a crucial role in determining the thrust produced. It is the velocity at which the mass ejected leaves the rocket, directly affecting the efficiency and power of the propulsion system. Rocket engineers carefully calculate exhaust velocity to ensure optimal performance of the rocket engine.In our example, the exhaust velocity is \( 2000 \, \mathrm{m/s} \), which affects how much mass must be ejected to achieve sufficient thrust for the desired acceleration. The connection between thrust, mass ejection, and exhaust velocity is given by the formula \( F = \Delta m \cdot v_e \). Here, \( v_e \) is the key variable ensuring that the ejected mass results in the right amount of force to propel the rocket. Understanding exhaust velocity helps in designing engines capable of creating enough force while optimizing fuel efficiency.