How can we inpretate for falling many sphere?

In summary, the conversation discusses the topic of probability and the use of exponential function to express it. The speaker is wondering how to show the exponential nature of probability through an experiment involving a speaker, function generator, and a BB. They also mention the randomness of the BB's movement and the need for a theoretical background before designing an experiment. The concept of exponential dependence is explained, with an example of a well-known coin-tossing experiment that follows an exponential pattern. The speaker is encouraged to have a reason to believe in the exponential relationship before conducting an experiment.
  • #1
Choi Si Youn
7
0
Hi, in this thread I want to discuss some problem that I wonder.

I'm interested in "Probablilty"

The first wondering is How can I express that probability follow an exponential.
I'm wonder why probability expess in exponential?
and Can I show it, if I experiment something, that graph is exponential.

So I am setting some kind of experiment.
Express that experiment, I use speaker that I used for audio.
and I use function generator and audio amp.


I give a signal from function generator to speaker with surronded by sylindrical plastic cover.

and I put a BB(plastic sphere) into a speaker.


then I turn on the audio and function generator and check the height of jump of BB.


is it ok for my purpose?

and I still don't know that why probability follow the exponential?

and Why that BB jump randomly? the function generator and the audio give a same energy.



I can't apply between the textbook when I studied and the actual.
 
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  • #2
Before you design an experiment, you need a theoretical background to guide your experimental procedure. Here, I assume you measured the height of the jumping BB and that's your dependent variable. However, you do not mention what your independent variable was. What did you vary in a controlled way as you measured the height? You mentioned a graph. Did you make one? If so, what was on the x-axis? Besides all this, what reason do you have to believe that the height of the BB has an exponential dependence on your independent variable, whatever that is?

The hallmark of the exponential dependence of a dependent variable ##y## on an independent variable ##x## is that the rate of change of ##y## with respect to ##x## is proportional to ##y##. Mathematically, this is expressed as
$$\frac{dy}{dx}=ky.$$Here, ##k## is the constant of proportionality. Before you design an experiment to test the exponential dependence, you need to have a reason to believe that your dependent and independent variables more or less obey this relation. For example, a well known and easy to do exponential experiment is based on coin-tossing. Here's what you do

Start with a large number of identical coins, say 100.
Place them all "heads" up. The number of heads is your dependent variable ##y## and the number of tosses is your independent variable ##x##. Your first data point {x,y} is {0, 100}. Now toss the coins up in the air. Say you get 52 heads and 48 tails. Your second data point is {1,52}. Remove the 48 tails and toss the remaining 52 heads. Your third data point could be {2,23}. Remove the coins landing as tails and so on until there are no heads left. Plot number of heads vs. number of tosses; the plot should be an exponential.

Why do I think it should be an exponential? Because I believe that there is a 50-50 probability that a coin will land heads or tails. This means that the number of coins that land heads per toss (##dy/dx##) is (more or less) half the number of the heads that I am tossing (##k=1/2##). See how it works?
 

1. How does the rate of falling spheres change with mass?

According to Newton's Second Law of Motion, the acceleration of a falling object is directly proportional to its mass. This means that the heavier the sphere, the faster it will fall due to the force of gravity.

2. What factors affect the trajectory of a falling sphere?

The primary factor that affects the trajectory of a falling sphere is air resistance. The more aerodynamic the shape of the sphere, the less air resistance it will experience, resulting in a more direct fall. Other factors such as wind and external forces can also impact the trajectory.

3. How can we measure the velocity of a falling sphere?

The velocity of a falling sphere can be measured using a variety of methods such as high-speed cameras, radar guns, or motion sensors. These tools can track the movement of the sphere and calculate its velocity based on the time it takes to fall a certain distance.

4. What is the equation for calculating the acceleration of a falling sphere?

The equation for calculating the acceleration of a falling sphere is a = g - (F_air/m), where a is the acceleration, g is the acceleration due to gravity (9.8 m/s^2), F_air is the force of air resistance, and m is the mass of the sphere. This equation takes into account the effect of air resistance on the acceleration of the sphere.

5. Can the shape of a falling sphere affect its acceleration?

Yes, the shape of a falling sphere can affect its acceleration. A more streamlined and aerodynamic shape will experience less air resistance, resulting in a faster acceleration. On the other hand, a more irregular or bulky shape will experience more air resistance and have a slower acceleration.

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