# Two-Stage Rocket: Find Speed w/ Fuel Exhaustion & v_ex Req'd for 6 km/s

• faller217
In summary, the conversation is about a two-stage rocket with specific masses of fuel and a constant relative speed of ejected material. It discusses the speed of a single-stage rocket with the same total mass, the speed when the first stage's fuel is exhausted, the final speed of the second stage, and the required value of v_ex for a speed of 6.00 km/s. The relevant equations for this type of problem are not provided and the person is reminded to show their work instead of asking for direct assistance.
faller217
I was hoping someone could help me with this homework problem that has me stumped. Any assistance would be appreciated.

Suppose the first stage of a two-stage rocket has total mass 1.30×10^4 kg, of which 1.10×10^4 kg is fuel. The total mass of the second stage is 1000 kg, of which 700 kg is fuel. Assume that the relative speed v_ex of ejected material is constant, and ignore any effect of gravity. (The effect of gravity is small during the firing period if the rate of fuel consumption is large.)

A) Suppose the entire fuel supply carried by the two-stage rocket is utilized in a single-stage rocket with the same total mass of 1.40×10^4 kg. In terms of v_ex, what is the speed of the rocket, starting from rest, when its fuel is exhausted?

B)For the two-stage rocket, what is the speed when the fuel of the first stage is exhausted if the first stage carries the second stage with it to this point? This speed then becomes the initial speed of the second stage. At this point, the second stage separates from the first stage.

C)What is the final speed of the second stage?

D)What value of v_ex is required to give the second stage of the rocket a speed of 6.00 km/s?

We don't do your homework for you. Please show some work. What are the relevant equations for this kind of problem?

A) To find the speed of the single-stage rocket, we can use the conservation of momentum equation: mv = (m + dm)(v + dv), where m is the initial mass of the rocket, dm is the change in mass due to fuel consumption, and v and dv are the initial and final velocities, respectively. In this case, v is 0 since the rocket starts from rest, and dm is the total fuel mass of 2.10×10^4 kg. Solving for v, we get v = v_ex ln(m / (m + dm)). Plugging in the given values, we get v = v_ex ln(1.30×10^4 / 1.40×10^4) = -0.074v_ex. Therefore, the speed of the single-stage rocket when its fuel is exhausted is -0.074v_ex.

B) When the fuel of the first stage is exhausted, the mass of the rocket becomes 2.70×10^3 kg (1.40×10^4 kg - 1.10×10^4 kg - 700 kg). Using the same conservation of momentum equation, we get v = v_ex ln(m / (m + dm)) = v_ex ln(2.70×10^3 / 1.40×10^4) = -0.993v_ex. This becomes the initial speed of the second stage when it separates from the first stage.

C) To find the final speed of the second stage, we can again use the conservation of momentum equation. The initial mass of the second stage is 1000 kg and the final mass is 300 kg (700 kg - 400 kg). The initial velocity is -0.993v_ex and the final velocity is v. Solving for v, we get v = 3.31v_ex. Therefore, the final speed of the second stage is 3.31v_ex.

D) To find the value of v_ex required to give the second stage a speed of 6.00 km/s, we can use the final speed equation from part C and set it equal to 6.00 km/s. This gives us 6.00 km/s = 3.31v_ex, so v_ex = 1.81 km/s. Therefore, the required value of v_ex is 1.81 km/s to give the second stage a speed of

## What is a two-stage rocket?

A two-stage rocket is a type of rocket that uses two separate stages or sections to reach higher altitudes and velocities. The first stage is usually the larger section and contains the main engines and most of the fuel. Once the first stage has burned through its fuel, it separates from the rocket and the second stage takes over to continue the ascent.

## How does a two-stage rocket work?

A two-stage rocket works by using the force of combustion to propel itself upwards. In the first stage, the fuel and oxidizer are ignited, creating a controlled explosion that pushes the rocket upwards. Once the fuel is exhausted, the first stage is jettisoned and the second stage takes over. The second stage can either use a different type of fuel or the same fuel in a more efficient way to continue the ascent.

## What is the purpose of finding the speed with fuel exhaustion for a two-stage rocket?

The purpose of finding the speed with fuel exhaustion for a two-stage rocket is to determine the maximum velocity that can be achieved before the first stage runs out of fuel. This information is crucial for planning the trajectory and ensuring that the rocket reaches its desired destination.

## Why is it important to know the v_ex required for a two-stage rocket to reach a velocity of 6 km/s?

Knowing the v_ex (exhaust velocity) required for a two-stage rocket to reach a velocity of 6 km/s is important because it helps in the design and construction of the rocket. The v_ex is a measure of the efficiency of the rocket's engines, and knowing this value allows engineers to determine the type and amount of fuel needed for the rocket to reach the desired velocity.

## What factors affect the v_ex required for a two-stage rocket?

The v_ex required for a two-stage rocket can be affected by various factors, including the weight and design of the rocket, the type of fuel used, and the desired velocity. Other external factors such as atmospheric conditions and gravitational pull can also influence the v_ex required for a two-stage rocket to reach a certain velocity.

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