Answer to Question #95078 in Mechanics | Relativity for Najia

Question

B. Now imagine that you are living in a world where free-falling objects have an
upwards jerk (i.e. a negative jerk if down is positive). In other words, they
accelerate downwards, but the downwards acceleration gets smaller and smaller
with each beat.
i. Design a pattern of bolt positions that might produce a steady rhythm in this
imaginary world. Think hard. This is hard.

BIG Hint: to do this, you will have to pick some arbitrary, large downwards
acceleration to start out with, and some small constant upwards jerk
(amount the acceleration shrinks by for each beat). But remember: jerk is a
change in acceleration, not a change in velocity!
Show ALL work relating to your pattern.

ii. Draw both a position and a displacement diagram for your pattern.

iii. Using math, words, & diagrams (as necessary), explain why your pattern
should produce a steady rhythm in the imaginary world where free-fall
acceleration is downward but jerk is upward: i.e. justify your solution.

Answer 1

i. Assume that we have some initial positive acceleration $a_0$. Then write velocity, acceleration and jerk as derivatives:

$$v=\frac{dx}{dt},\space\space a=\frac{dv}{dt}\space\space j=\frac{da}{dt}.$$

Hence by integration:

$$a=a_0-jt,\\ v=v_0+a_0t-\frac{1}{2}jt^2,\\ x=x_0+v_0t+\frac{1}{2}a_0t^2-\frac{1}{6}jt^3.$$

The constants $x_0, v_0, a_0$, obtained during the integration process, represent initial position, velocity and acceleration. We can assume that initial velocity and position are 0, but we need some initial acceleration, let it be 50 m/s/s, and some jerk, let it be 12. Substitute this:

$$x=25t^2-2t^3.$$

The points are:

$$t:\space0\space\space1\space\space\space2\space\space\space\space3\space\space\space\space4\space\space\space\space\space5\space\space\space\space\space6\space\space\space\space\space7\space\space\space\space\space8\space\space\space\space\space9\space\space\space\space10\\ x:\space0\space23\space84\space171\space272\space375\space468\space539\space576\space567\space500$$

We can notice that seconds 0 to 6 can represent the steady rhythm, because the position increases with time..

ii. The diagrams are below:

From the displacement diagram we see that initially the body is accelerating, then decelerating, then again it is accelerating downwards.

iii. From the condition (acceleration is downward but jerk is upward) and from the equation obtained in part (i) we see that initially, with small values of time, the coefficient before time squared, i.e. 50, overcomes the second negative member. However, with large t, the part with time cubed becomes significantly larger. On the other hand, we took beats each equal to 1 second, that is why the rate of change of acceleration was steady.

According to Wikipedia: "In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies." In part (i) we obtained two different graphs ("frequencies", i.e the one is $50t^2$ and the other is $-2t^3$), that is why, when they interfered at the time interval between 7 and 9 seconds, we "perceived" that interference, i.e. acceleration, deceleration, and acceleration again. In case of sounds it would be up-down-up the volume.