b. A rocket with a total mass of 330,000 kg when fully loaded burns all 280,000 kg of fuel in 250 s. The engines generate 4.1 MN of thrust. What is this rocket's speed at the instant all the fuel has been burned if it is launched in deep space? If it is launched vertically from the earth?

Verified step by step guidance

1. Step 1: Calculate the exhaust velocity of the rocket using the thrust and the mass flow rate. The mass flow rate can be found by dividing the total mass of the fuel by the burn time. Use the formula for thrust, \( F = \dot{m} v_e \), where \( F \) is the thrust, \( \dot{m} \) is the mass flow rate, and \( v_e \) is the exhaust velocity.
2. Step 2: Apply the rocket equation, \( \Delta v = v_e \ln\left(\frac{m_i}{m_f}\right) \), where \( \Delta v \) is the change in velocity, \( m_i \) is the initial total mass (including fuel), and \( m_f \) is the final total mass (after fuel is burned). Calculate the natural logarithm of the ratio of the initial mass to the final mass.
3. Step 3: Multiply the exhaust velocity by the natural logarithm value obtained in the previous step to find the change in velocity (\( \Delta v \)) of the rocket.
4. Step 4: For the scenario in deep space, assume there are no external forces acting on the rocket (like gravity or atmospheric drag), so the speed of the rocket at the instant all fuel is burned is simply the change in velocity calculated.
5. Step 5: For the scenario of launching vertically from Earth, consider the effect of gravity. Subtract the product of gravitational acceleration (approximately 9.81 m/s^2) and the burn time from the change in velocity calculated to account for the loss in speed due to gravity.

Key Concepts

Thrust and Newton's Second Law

Thrust is the force exerted by a rocket's engines to propel it forward, measured in newtons (N). According to Newton's Second Law, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle is crucial for understanding how the rocket's thrust affects its motion and speed during fuel consumption.

Rocket Equation (Tsiolkovsky's Rocket Equation)

The Tsiolkovsky Rocket Equation relates the velocity of a rocket to its mass and the effective exhaust velocity of the propellant. It is expressed as Δv = ve * ln(m0/mf), where Δv is the change in velocity, ve is the effective exhaust velocity, m0 is the initial mass, and mf is the final mass after fuel is burned. This equation is essential for calculating the final speed of the rocket after all fuel has been expended.

Gravitational Force and Launch Conditions

When launching a rocket from Earth, it must overcome gravitational force, which affects its acceleration and speed. The gravitational force can be calculated using F = mg, where m is the mass of the rocket and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). In deep space, the rocket experiences negligible gravitational pull, allowing it to achieve higher speeds compared to a launch from Earth.