Solving Physics Questions: Muons, Speakers & Sound Intensity

[Pseudo Statistic]

Hi.
I'm having a bit of problems with this question:

A group of muons is observed to pass a balloon at a height of 480 meters above the Earth's surface. The muons move at a speed of 0.8c straight down towards the Earth's surface.
a) Calculate the time for the muons to travel from the balloon to the Earth's surface as determined by an observer on the Earth.
b) Calculate the time for the muons to travel from the balloon to the Earth's surface as determined by an observer moving with the muons. Explain your reasoning.

I have no clue how to do them because that "c" is throwing me off-- like, would they be moving with that constant velocity or will gravity be accelerating them further up to c and then they'll be moving with constant velocity, or is this some relativistic thing?

Also, let's say you had two speakers, a distance d apart and both a distance L away from an axis; i.e. something like this:
http://www.brokendream.net/xh4/physpic.JPG
Assume B, B and the speakers all lie on a vertical line and both speakers vibrate in phase and emit sound waves of equal amplitude and wavelength... and assuming d << L.
a) Describe how sound intensity I varies as a function of position x along the line segment OA. Sketch the graph of this function on an axes.

b) Assuming wavelength << d sketch a graph of the sound intensity I as a function of position y along the y axis.

c) Assume that d = 2m and the speed of sound = 360 m/s. Find the lowest speaker frequency which will yield the minimum sound intensity along the line BB'.

I wouldn't want someone to do these questions for me, however, I do want some pointers as to what fundamental equations I should be looking at and what topics to study.

I appreciate any help I get! THANKS!

 

[Hootenanny]

For question (a), if you assume they are moving at a constant velocity (the given 0.8c) you will have to take into account the time dilation effect because the muon is traveling relatively close to the spead of light. The formula for which is;

$$T = \frac{T_{0}}{\sqrt{1-\frac{v^2}{c^2}}}}$$

You might want to take a look at this page;
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html#c2

(b) is along the same lines. HINT: Your answer to a and b should be different.
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For the sound questions, you need to think about interference patterns, conditions for constructive / destructive interference etc.

Regards,
-Hoot

 

[kyle8921]

Hello there,

I can understand your confusion with these questions. Let's break them down and see if we can make sense of them.

For the first question about muons, the "c" refers to the speed of light in a vacuum, which is a constant value of 299,792,458 meters per second. This is a very fast speed and is often used in physics calculations involving relativity. In this case, the muons are moving at 0.8 times the speed of light, which is still very fast but not quite as fast as the speed of light. So, we can assume that the muons are moving at a constant velocity and not being accelerated by gravity.

To calculate the time for the muons to travel from the balloon to the Earth's surface, we can use the equation d = vt, where d is the distance traveled, v is the velocity, and t is the time taken. In this case, we know the distance (480 meters) and the velocity (0.8c), so we can rearrange the equation to solve for t. This will give us the time for an observer on Earth.

For an observer moving with the muons, we need to take into account the concept of time dilation, which is a consequence of Einstein's theory of relativity. Basically, time appears to pass slower for an object moving at a high velocity compared to an object at rest. So, for the observer moving with the muons, the time will appear to pass slower, and therefore the time taken for the muons to travel from the balloon to the Earth's surface will be longer.

Moving on to the second question about sound intensity and speakers, there are a few key equations and concepts to consider. Sound intensity is defined as the power per unit area and is measured in watts per square meter. It is directly proportional to the amplitude squared of the sound wave. So, as the amplitude increases, the sound intensity also increases.

a) For the first part of the question, we can use the equation for sound intensity, I = P/A, where P is the power and A is the area. In this case, the power is constant because the speakers are vibrating in phase and emitting sound waves of equal amplitude. The area, however, will vary as we move along the line segment OA. As the distance from the speakers increases, the area will also increase, resulting in a decrease in sound intensity. So,