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Computer drawing of a ball showing simple translation
 and the definitions of average velocity and acceleration.

We live in a world that is defined by three spatial dimensions and one time dimension. Objects move within this domain in two ways. An object translates, or changes location, from one point to another. And an object rotates, or changes its orientation. In general, the motion of an object involves both translation in all three directions, and rotation about three principle axes.

On this page we will consider only the translation of an object. We can specify the location of the object 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 object is at point "0", with coordinates x0, y0, and z0 at time t0. In general, the object moves through the domain until at some later time t1 the object is at point "1" with coordinates x1, y1, and z1. We can specify the displacement - d in each coordinate direction by the difference in coordinate from point "0" to point "1". The x-displacement equals (x1 - x0), the y-displacement equals (y1 - y0), and the z-displacement equals (z1 - z0). On this web page, for simplicity, we are only going to consider the x coordinate. The displacement is then given by:

d = x1 - x0

Displacement is a vector quantity, which means that displacement has a size and a direction associated with it. The direction is from point "0" to point "1". The individual x-, y-, and z-displacements are the components of the displacement vector in the coordinate directions. All of the quantities derived from the displacement are also vector quantities.

The velocity -V of the object through the domain is the change of the location with respect to time. In the X - direction, the average velocity is the displacement divided by the time interval:

V = (x1 - x0) / (t1 - t0)

This is just an average velocity and the object might speed up and slow down between points "0" and "1". At any instant, the object 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;

V = dx / dt

where the symbol d / dt is the differential from calculus. (For younger students, you have just had your first lesson in calculus! .. it wasn't so bad, was it? If this doesn't make sense, don't worry and just keep reading .. it will make sense in a few years.)

When we initially specify the location of our object with x0, y0, z0, and t0 coordinates, we should also specify an initial instantaneous velocity V0. Likewise at the final position x1, y1, z1, and t1, the velocity changes to a velocity V1. Again, for simplicity, we are considering only the x-component of the velocity. In reality, the velocity changes in all three directions. Velocity is a vector quantity and has both a magnitude and a direction. The direction is the same as the the displacement direction from which we defined the velocity.

The acceleration (a) of the object through the domain is the change of the velocity with respect to time. In the X - direction, the average acceleration is the change in velocity divided by the time interval:

a = (V1 - V0) / (t1 - t0)

As with the velocity, this is only an average acceleration. At any instant, the object 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:

a = dv / dt

From Newton's second law of motion, we know that forces on an object produce accelerations. If we can determine the forces on an object, and how the forces change with time, we can use the equations presented on this slide to determine the acceleration, velocity, displacement and, therefore, location of the object as a function of time. Aeronautical engineers use this information to predict the motion of an aircraft, or the flight of a rocket.


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Editor: Nancy Hall
NASA Official: Nancy Hall
Last Updated: May 05 2015

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