3.4    Acceleration and Energy

Velocity is a vector quantity, consisting of both magnitude and direction.  Acceleration is any change in velocity, whether in the magnitude or the direction.  By the second law of motion, acceleration requires the application of a force.  But the application of a force over any displacement requires work.  The work done in accelerating a body is manifested as a change in its kinetic energy:




 W  is the work required to accelerate the body.

 delta Ek  is the change in the kinetic energy of the body.

 m  is the mass of the body.

 F  is the accelerating force.

 R, R_dot, R_dot_dot  are the position, velocity, and acceleration of the body.

 R, V, A  are the magnitudes of R, R_dot, R_dot_dot.

 Θ  is the angle between R_dot and R_dot_dot.

Figure 3.5

Figure 3.5:  Linear acceleration.

Figure 3.6

Figure 3.6:  Centripetal acceleration.

If R_dot_dot is parallel to R_dot, then the acceleration is linear and affects only the magnitude of R_dotΘ is zero, cos(Θ) is one, V is a function of time, and A is dV/dt.  Then equation 3.7 yields:



On the other hand, if R_dot_dot is perpendicular to R_dot, then the acceleration is centripetal and affects only the direction of R_dotΘ is π/2, cos(Θ) is zero, and V is constant.  Then equation 3.7 yields:



Equation 3.6 shows the relationship between acceleration and apparent weight, and equations 3.7 - 3.9 show the energy costs associated with acceleration.  Using these equations, we can consider the relative merits of various strategies for producing the sensation of weight in space.

Equation 3.8 gives some indication of the enormous energy costs associated with constant linear acceleration.  The principle of the conservation of energy dictates that the kinetic energy of a body can be increased only through an equivalent decrease in other energy stores - for example:  gravitational, chemical, electrical, nuclear, or the kinetic energy of another body.  Thus, the linear acceleration needed to produce an appreciable level of apparent weight can be sustained for only a limited amount of time, and with foreseeable technology that limit is rather low.  Furthermore, linear acceleration precludes a body from staying put in any finite region of space, making it incompatible with the concept of orbiting space stations.  On the other hand, if the necessary technology could be developed, constant linear acceleration might be compatible with interplanetary or interstellar travel.  For example, a trip to Venus might be completed in as little as 35 hours [10].  Furthermore, since the acceleration is constant in both magnitude and direction, it is a nearly perfect surrogate for natural gravity, as Einstein showed in his thought experiment.

Equation 3.9 shows that there is no energy cost associated with centripetal acceleration.  A centripetal force is required to accelerate the body, but because this force is always perpendicular to the velocity, it does not act through any displacement; therefore, it performs no work and consumes no energy.  Constant-magnitude centripetal acceleration is of particular interest: in this case, the body follows a circular trajectory.  (This will be proven in the next section.)  The required centripetal force is easily provided by connecting the body to one or more counterbalancing bodies by means of a tensile structure; the entire system then rotates about its combined center of mass.  Some initial energy input is required to start the system rotating, but once it is started, it will rotate forever with no further energy expense, unless some outside force (such as friction) acts to stop it.  Furthermore, centripetal acceleration is compatible with the concept of orbiting space stations as well as with interplanetary or interstellar travel.  Equation 3.6 shows that gravitational acceleration does not contribute to apparent weight; therefore, the entire rotating system can be placed in any coasting free-fall trajectory - Earth orbital or otherwise - without affecting the apparent weights of its constituent elements.  On the other hand, centripetal acceleration is a less-than-perfect replacement for natural gravity: although its magnitude may be constant, its direction is not.  The constantly-changing acceleration vector has distinctly uncommon effects on bodies moving within the rotating system.

Constant linear acceleration appears unachievable in the near future, has limited application, and will be a nearly perfect substitute for natural gravity if the necessary technology is ever developed.  For all of these reasons, the focus of this research is on centripetal acceleration (and its side effects), which is easily achievable, has wider application, and leads to some interesting design problems.