Coaster Dynamics: Physics Primer Chapter 6



Coaster Dynamics
Physics Primer

Chapter 6

Motion in 3 Dimensions

The derivation of the 3-dimensional velocity and acceleration equations is accomplished by the direct extension of the analysis of 2D motion in the previous chapter into the 3'rd dimension, z.

While straightforward in theory, this procedure isn't necessarily easy. The equations and notation can become rather cumbersome, and 3D calculations get quite tedious to perform due to the large amount of "bookkeeping" required. For this reason, until the advent of computers, 3D calculations were only within the realm of math wizards. But what the heck, that's what computers were invented for... right?

3D Particle Velocity

Leaving the details for the reader to derive, the average velocity of a particle in 3 dimensions is equal to:

(6-1)         v = (change in position)/(change in time) = s/t.

    (6-2)         vx = (x2 - x1)/(t2 - t1)
    (6-3)         vy = (y2 - y1)/(t2 - t1)
    (6-4)         vz = (z2 - z1)/(t2 - t1)

    (6-5)         v = xvx + yvy + zvz.

The average velocity components can be calculated analytically from equations (6-3) through (6-5), or by the graphical method discussed in the previous chapter.

Also, the average velocity components will approach the true instantaneous velocities as the time interval for measurement is made smaller and smaller.

3D Particle Acceleration

The average acceleration of a particle in 3 dimensions is equal to:

(6-6)         a = (change in velocity)/(change in time) = v/t.

    (6-7)           ax = (vx2 - vx1)/(t2 - t1)
    (6-8)           ay = (vy2 - vy1)/(t2 - t1)
    (6-9)           az = (vz2 - vz1)/(t2 - t1)

    (6-10)         a = xax + yay + zaz.

The average acceleration components can be calculated analytically from equations (6-7) through (6-10), or by the graphical method discussed in the previous chapter.

And once again, the average acceleration components will approach the true instantaneous accelerations as the time interval for measurement is made smaller and smaller.

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