# How Does Rocket Ejection Mass Affect Its Velocity in Space?

• Father_Ing
In summary, the conversation discusses the equation for the final velocity of a rocket moving in space, which involves the conservation of momentum. The participants also question whether the velocity can increase or decrease after a small amount of time, with one participant suggesting that by taking the limit of dm->0, the velocity remains unchanged. However, it is pointed out that this is not a valid approach as it ignores higher order contributions. Ultimately, it is concluded that the velocity neither increases nor decreases, but there may be errors in the method used to reach this conclusion.
Father_Ing
Homework Statement
-
Relevant Equations
momentum conservation
Consider a rocket with mass ##m## in space is going to move forward. In order to do so, it needs to eject mass backwards. Let the mass that is ejected has velocity ##u## relative to the rocket. What is the equation for the final velocity?

It is said that after ##dt## second, the rocket will have mass ##m-dm##, and velocity ##v+dv##.But, isn't it also possible for the speed to increase in high sum after a very small amount of time, or even, not changing at all?

And I tried to find this out by using conservation of momentum.
Let ##v'## be the rocket's speed after ##dt## second, and ##v## is the initial speed.
$$mv = dm(v-u)+(m-dm)v'$$
$$v'=\frac {(m-dm)v +udm}{m-dm}$$

$$v'=v+\frac {udm}{m-dm}$$
Since ##dm## is small, we can take the limit of dm->0. Therefore,$$v'= v$$
It can be concluded that the velocity neither increase nor decrease.

But, I searched about this matter in the internet, and they said that ##v'## is ##v+dv##. Are there any mistakes in my method?

Father_Ing said:
Homework Statement:: -
Relevant Equations:: momentum conservation

Since dm is small, we can take the limit of dm->0.
No you cannot. You are looking for dv, the small change in velocity when you eject mass dm. Obviously, if you do not eject any mass at all (ie, take the limit dm -> 0), then velocity does not change.

You want to find the change dv while ignoring higher order contributions (ie, ##dm^2## etc).

## 1. How does a rocket move in space?

A rocket moves in space by expelling gas at high speeds through its engine. This action creates a force in the opposite direction, propelling the rocket forward.

## 2. What is the difference between rocket motion in space and on Earth?

Rocket motion in space is different from on Earth because there is no air resistance or gravity in space. This means that rockets can travel at much higher speeds and in any direction without being affected by external forces.

## 3. How do rockets change direction in space?

Rockets change direction in space by using small thrusters or by rotating the entire rocket. This allows them to adjust their trajectory and change their direction of travel.

## 4. How do rockets navigate in space?

Rockets navigate in space by using a combination of sensors, such as gyroscopes and star trackers, and computer systems. These sensors help the rocket determine its position and velocity, and the computer calculates the necessary adjustments for navigation.

## 5. What is the maximum speed a rocket can reach in space?

The maximum speed a rocket can reach in space is determined by its engine's capabilities and the amount of fuel it carries. Currently, the fastest speed achieved by a rocket in space is about 25,000 miles per hour, achieved by the New Horizons spacecraft on its journey to Pluto.

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