# Rocket ascending in Earth's gravity

• kraigandrews
In summary, the problem discusses a rocket ascending in Earth's gravitational field, using exhaust with constant speed u and experiencing air resistance with a force mbv. The velocity of the rocket as a function of time is given by v(t)=(1/b)e^((-uγ/b)t)-(g/b). The constant γ is not specified, so the solution must be expressed in terms of it.
kraigandrews

## Homework Statement

A rocket ascends from rest in Earth's gravitational field, by ejecting exhaust with constant speed u. Assume that the rate at which mass is expelled is given by dm/dt = −γm where m is the instantaneous mass of the rocket and γ is a constant; and that the rocket is retarded by air resistance with a force mbv where b is a constant.
Determine the velocity of the rocket as a function of time. Here is a hint: The terminal velocity is ( γu−g )/b.

Calculate the time when the velocity is one-half of the terminal velocity.
Data: u = 31.9 m/s; b = 1.2 s−1.

dp/dt=F=m(dv/dt)

## The Attempt at a Solution

I get dv=-udm-(g+bv)dt; dm=-γm
so dv=uγ-(g+bv)dt

solving for v:
v(t)=(1/b)e^((-uγ/b)t)-(g/b)

the problem I am running into is what is gamma, because I have no inital condition to apply, and I'm fairly sure the solution to the diff eq is correct.

gamma is just a constant, it doesn't really matter what it is, your answers will probably just have to be expressed in terms of it.

## 1. How does a rocket overcome Earth's gravity?

A rocket overcomes Earth's gravity by using a powerful engine to generate thrust, which propels the rocket upwards. As the rocket moves upwards, it gains speed and eventually breaks free from the pull of Earth's gravity.

## 2. What is the role of gravity during a rocket launch?

Gravity plays a crucial role during a rocket launch. It provides the initial force that propels the rocket upwards, and as the rocket gains altitude, gravity continues to pull it back towards Earth. This results in an arcing trajectory that eventually leads to the rocket entering into orbit around the Earth.

## 3. How does the weight of a rocket affect its ascent in Earth's gravity?

The weight of a rocket is directly related to the amount of thrust needed to overcome Earth's gravity. The heavier the rocket, the more thrust is required to lift it off the ground. This is why rockets are designed to be as lightweight as possible while still being able to carry the necessary payload.

## 4. Can a rocket escape Earth's gravity entirely?

Yes, a rocket can escape Earth's gravity entirely by achieving a speed known as escape velocity. This is the minimum speed required for an object to break free from the pull of Earth's gravity and travel into space. Once a rocket reaches escape velocity, it can travel beyond Earth's atmosphere and into the depths of space.

## 5. How does the shape of a rocket affect its ascent in Earth's gravity?

The shape of a rocket is designed to minimize air resistance and maximize thrust, which helps the rocket overcome Earth's gravity more efficiently. A streamlined shape allows the rocket to move through the air more easily, reducing drag and allowing it to reach higher speeds. Additionally, the shape of the rocket's fins and nose cone can also affect its stability and overall performance during ascent.

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