Question

A. Imagine that your lab group (which may or may not be taking physics in an alternate reality) creates a string with screws at positions 1m, 2m, 3m, 4m, and 5m. You drop this string from a height of 5m, just as the lab instructions tell you to do, and you notice that the time intervals between screws hitting the cookie sheet get larger with each screw. The first two impacts are close together, then a little farther apart, then farther, and farther. Assume that all your measurements and observations are more or less accurate and that you performed your drop correctly. What would you conclude from your data about the behavior of objects in free fall?

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

A. From this data, we can conclude that there is at least an acceleration in the downwards direction because if not, then then either the velocity is constant, or there is an upward acceleration. If the former, then equal displacement spacing should take equal time to cover, and hence the pattern should be linear, which is not the case. If the former, then since we are starting with an initial velocity of 0, the body should move upwards, which does not happen. And hence the conclusion.

B. (i) Consider a system with positive acceleration and negative jerk . Then, using the fact that , we get:

where are the initial position, velocity and acceleration respectively. Let and . But, the acceleration is constant, so let . Then:

Now, for the sake of a concrete model, let and . Thus .

0	0
1	14
2	52
3	108
4	176
5	250

Thus if we have a set of markings , we can set up a steady rhythm that counts up to 5.

(ii) The above is the displacement-time graph, with time on the x-axis and displacement on the y-axis. As for the position time graph, The below graph shows that (the markings start from the bottom and go all the way up) :

(c) Our pattern follows a steady rhythm because they were precisely calculated at beats of 1, i.e. the time interval between two consecutive markings is 1, and hence they would produce a steady beat.