Calculating Thrust and Mass Requirements for a Rocket with Different Loads

The problem presented involves determining the necessary thrust and mass of a rocket to achieve a specific height with a given load. This article explores the calculations involved and clarifies the assumptions needed to reach a coherent solution.

Understanding the Given Data

The initial problem sets up a scenario where a projectile with a mass of 3130 kg (including a payload of 400 kg) is lifted to a height of 1286 km in 680 seconds with an upward thrust of 58100 N. The key forces at play are the upward thrust and the downward gravitational force.

Force Analysis

The upward thrust force, Ft, is 58100 N. The gravitational force, P, acting on the total mass, Mtotal, is given by:

P Mtotal * g 3130 400 10 35300 N

The net force acting on the projectile, and therefore its acceleration, is:

a (Ft - P) / Mtotal (58100 - 35300) / 3530 6.46 m/s2

Kinematic Analysis

Using the equations of kinematics, we find the initial velocity, Vo, which must satisfy the condition to reach 1286 km (1286,000 m) in 680 seconds.

V Vo - a * t, and Y Vo * t - (1/2)a * t2

Given Y 1286,000 m and t 680 s:

Vo 4.08 m/s

This solution raises a question because with the given values, the projectile should peak at around 0.63 seconds and fall back after 680 seconds, which is inconsistent.

Clarifying Assumptions

For a more accurate calculation, we need to clarify a few assumptions:

The 3130 kg is the total mass of the rocket including fuel and payload, with the 400 kg being part of the payload. The 1286 km is the final height obtained after the fuel is exhausted, and the rocket and payload eventually come to a stop due to air resistance and fall back to the ground.

Calculations

Given that the acceleration is directly dependent on the required thrust, we can use the formula:

acceleration force / mass

Original Calculation

To lift the original projectile:

acceleration 58100 N / (3130 400) kg 16.92 m/s2

New Calculation for 3000 kg Load

To lift a new load of 3000 kg to the same height with the same acceleration:

thrust mass * acceleration 3000 kg * 16.92 m/s2 50760 N

mass needed thrust / acceleration 50760 N / 16.92 m/s2 3000 kg

Thus, a rocket with a total mass of 3000 kg (without considering fuel) and a thrust of 50760 N is required.

Understanding Thrust and Mass Requirements

The thrust required to lift an object is the force needed to overcome gravity and provide the necessary acceleration. As the mass of the object increases, the required thrust also increases, leading to a larger rocket mass.

Given:

Thrust (T): The total force exerted by the rocket to overcome gravity and provide lift. Mass (m): The total mass of the rocket, including fuel and payload. Acceleration (a): The rate at which the rocket's velocity changes due to the thrust force.

These relationships are crucial in rocket design and can be summarized by the following equations:

Thrust mass x acceleration mass thrust / acceleration

Fuel Consumption and Rocket Design

When designing a rocket, the fuel consumption is a critical factor. The total mass of the rocket must be carefully managed to ensure that the required thrust is provided for the entire flight. Fuel not only adds to the initial mass of the rocket but is also consumed, reducing the overall mass over time.

Uncertainty in fuel consumption can lead to discrepancies in final rocket mass. Therefore:

Accurately estimate fuel consumption: The mass of the fuel should be factored into the initial mass of the rocket. Design thrust and acceleration goals: Determine the necessary thrust and acceleration to achieve the desired height in the specified time. Iterative design process: Refine the design through simulation and testing to ensure the rocket meets the performance requirements.

Conclusion

Calculating the thrust and mass requirements for a rocket with different loads involves a careful analysis of the forces acting on the rocket and the assumptions underlying the problem. Clear understanding and accurate assumptions are crucial for successful rocket design and mission planning.

Clarifying the given data and ensuring consistent application of physical principles can lead to more reliable and accurate designs.