Acceleration - Initial Condition

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A problem statement, regarding dynamics of a BODY, contains specification of the acceleration of the BODY at the instant of initiation of its event as a "given." Thus the author implies, by some physical means or manner, "at the commencing instant of an event" the acceleration of the BODY is determined. Below is a discussion of how, physically, that "initial condition" might be known.

Conditions and the "Pre-Event:" Regarding acceleration in general. If at an instant in time, a BODY has an acceleration (non-zero) then in the next instant the BODY moves. The BODY is not where it was an instant later. Further, from its initial location, should a BODY not change location for its event, then it had no acceleration as its initial conditon.

The "physical/mathematical" meaning of event is important. Our discussion addresses the "event" of a BODY. The event initiates at an instance in time with the BODY characteristics at that time - its "initial conditions." The event procedes in time (for its duration, epoch or era) then, abruptly the event terminates. Interestingly, to state the "initial condition" of a BODY, requires consideration of another event, specifically, the "end" of the BODY event immediately prior the event under investigation.

Cases: Below are descriptions of physical circumstances "prior to our event" that assure a known initial acceleration of our BODY for its event.

♦ BODY above Earth If a BODY is suspended above Earth by the tension in a "constraining cord," and the event is described to commence upon, sever of the cord - the initial acceleration is rightly stated as the surface of Earth value, g_o,E .

♦ Vehicle Accelerated In this case a mass hangs on a cord attached to the rear-view mirror of an auto. While stationary, the cord alignment is vertical. Moments later the auto is in motion. Sketch (a) shows a condition of the pendant during motion.

Calculate the acceleration of the car at the instant sketch (a) applies.

It is logical to model the pendant as a BODY and to apply Newton's 2nd Law. A first step to apply Newton's Method is to "identify a system" then to "isolate" it. This is a mental, pencil and paper activity. To proceed in application

The "free-body-diagram" provides a vector basis for writing Newton's 2nd Law for the mass. The forces are the tension of the cord and the force of gravity.

(1) The length of a radial arc

We write the cord tension in magnitude/direction form. The angle of the cord with the vertical is notated (for now) as a function of time.

(2) to "find" the number.

For our physical event, the angle of inclination, β, is seen to be constant in time.

(3) to "find" the number.

Next we write the acceleration in component form, remove "time-dependence" from the angle, β then scalar multiply the(3) by the unit vectors I, then K. Two scalar equations result to be solved:

(4) to "find" the number.

The "third" equation of Eqn-4, has A_z = 0 because by observation the elevation of the BODY does not change. The BODY does not move in the "z" direction that component of its acceleration has magnitude, zero.

(5) to "find" the number.