# Rocket Translation

## Dimensions

We live in world that is defined by three spatial dimensions and one time dimension. Objects can move within this domain in two ways. An object can **translate**, or change **location**, from one point to another. And an object can rotate or change its **attitude**. In general, the motion of an object involves both translation and rotation. The motion of a rocket is particularly complex because the rotations and translations are coupled together; a rotation affects the magnitude and direction of the forces which affect translations.

## Translation of Rockets

On this page we will consider only the translation of a rocket within our domain. We can specify the location of our rocket at any time **t** by specifying three coordinates **x, y,** and **z** on an orthogonal coordinate system. An orthogonal coordinate system has each of its coordinate directions perpendicular to all other coordinate directions. Initially, our rocket is at point “0”, with coordinates **x _{0}, y_{0},** and

**z**at time

_{0}**t**. In general, the rocket moves through the domain until at some later time

_{0}**t**the rocket is at point “1” with coordinates

_{1}**x**and

_{1}, y_{1},**z**. We can specify the

_{1}**displacement – d**in each coordinate direction by the difference in coordinate from point “0” to point “1”. The x-displacement equals (x

_{1}– x

_{0}), the y-displacement equals (y

_{1}– y

_{0}), and the z-displacement equals (z

_{1}– z

_{0}).

**On this page we only present displacement in the y-coordinate to help the student better understand the fundamentals of motion.**

\(\LARGE d=y_{1}-y_{0}\)

The total displacement is a vector quantity with the **x-, y-, **and **z-**displacements being the components of the displacement vector in the coordinate directions. All of the quantities derived from the displacement are also vector quantities.

## Velocity of the Rockets

The **velocity -V** of the rocket through the domain is the derivative of the displacement with respect to time. In the Y – direction, the average velocity is the displacement divided by the time interval:

\(\LARGE V=\frac{y_{1}-y_{0}}{t_{1}-t_{0}}\)

This is only an average velocity; the rocket could speed up and slow down inside the domain. At any instant, the rocket could have a velocity that is different than the average. If we shrink the time difference down to a very small (differential) size, we can define the instantaneous velocity to be the differential change in position divided by the differential change in time:

\(\LARGE V=\frac{\text{d}Y}{\text{d}t}\)

where the symbol **d / dt** is the differential from calculus. So when we initially specified the location of our aircraft with **x _{0}, y_{0}, z_{0},** and

**t**coordinates, we could also specify an initial instantaneous velocity

_{0}**V**. Likewise at the final position

_{0}**x**and

_{1}, y_{1}, z_{1},**t**, the velocity could change to some

_{1}**V1**. We are here considering only the y-component of the velocity. In reality, the rocket velocity changes in all three directions. Velocity is a vector quantity; it has both a magnitude and a direction.

## Acceleration of the Rockets

The **acceleration (a)** of the rocket through the domain is the derivative of the velocity with respect to time. In the Y – direction, the average acceleration is the change in velocity divided by the time interval:

\(\LARGE a=\frac{V_{1}-V_{0}}{t_{1}-t_{0}}\)

As with the velocity, this is only an average. At any instant, the rocket could have an acceleration that is different than the average. If we shrink the time difference down to a very small (differential) size, we can define the instantaneous acceleration to be the differential change in velocity divided by the differential change in time:

\(\LARGE a=\frac{\text{d}V}{\text{d}t}\)

From Newton’s second law of motion, we know that forces on an object produce accelerations. If we can determine the forces on a rocket, and how they change, we can use the equations presented on this slide to determine the location and velocity of the rocket as a function of the time.