Particle Distance/Velocity - Natural Logarithms

In summary, the problem asks for the total distance traveled by a particle, and includes terms for velocity. To find the answer, the problem asks for an infinite amount of time before the particle actually stops, and then determines when v0 (the initial velocity) equals zero. Using the information given, the answer is found to be s=0.5.
  • #1
Chase.
12
0

Homework Statement



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I'm going to post an image due to the complex syntax in the problem...

The Attempt at a Solution



I'm not going to lie... I really have no idea where to even begin with this problem. Because it says total distance traveled by the particle, I'm assuming that the total distance would be the distance equation multiplied by two; I'm not quite sure where velocity fits in however.

If I'm way off base, I'm still not looking for anyone to explicitly write out the answer for me. I just want to get headed in the right direction.
 
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  • #2
First thing you have to determine is when the particle stops. Look at that equation: v(t) = v0 e^(-at). Does v(t) ever actually equal zero?

You need an infinite amount of time before the particle actually stops. In practical terms, the particle will be found to have slowed practically to a stop in a measurable amount of time. However, for this theoretical exercise, t has to be infinitely large before v becomes zero.

At this point, e^(-at) vanishes (becomes zero). Can you now find the value of s at this point?
 
  • #3
Chase, the ball is back in your court. Show us what you have done, using the hints given by Courious3141 (this is, in fact, a very easy problem and Courious3141 gave very goo hint) or this thread will be deleted.
 
  • #4
HallsofIvy said:
Chase, the ball is back in your court. Show us what you have done, using the hints given by Courious3141 (this is, in fact, a very easy problem and Courious3141 gave very goo hint) or this thread will be deleted.
I'm not sure why this response was so aggressive.

Anyway, thanks Curious! I figured it out. I had initially considered the v0/a solution but discounted it because I didn't think I could input v0 in the submission box.
 
  • #5
Chase. said:
I'm not sure why this response was so aggressive.

Anyway, thanks Curious! I figured it out. I had initially considered the v0/a solution but discounted it because I didn't think I could input v0 in the submission box.

Glad you got it.:smile:
 

1. What is the significance of using natural logarithms to calculate particle distance/velocity?

Natural logarithms are used to model exponential growth or decay, which is often seen in particle distance and velocity. By taking the natural logarithm of a quantity, the resulting value is the exponent that would produce that quantity when used as the base of an exponential function. This allows for easier analysis and prediction of particle behavior.

2. How is particle distance/velocity affected by changes in natural logarithms?

Changes in natural logarithms can significantly impact particle distance and velocity. For example, if the natural logarithm of a particle's velocity increases, the particle's velocity will also increase exponentially. Similarly, if the natural logarithm of a particle's distance decreases, the particle will experience a rapid decrease in distance traveled.

3. Can natural logarithms be used to calculate the exact position and velocity of a particle?

No, natural logarithms can only be used to approximate the position and velocity of a particle. This is because particles follow a continuous path and can experience variations in velocity and position due to external factors such as air resistance and friction. However, natural logarithms can provide a good estimate of the particle's behavior.

4. Are there any limitations to using natural logarithms for particle distance/velocity calculations?

Yes, there are some limitations to using natural logarithms for particle distance/velocity calculations. For example, natural logarithms can only be used for continuous functions and may not accurately model abrupt changes in particle behavior. Additionally, they may not be suitable for particles that experience complex or chaotic motion.

5. How can natural logarithms be used to analyze particle behavior over time?

Natural logarithms can be used to create a graph of a particle's position or velocity over time. By plotting these values on a logarithmic scale, it is easier to see trends and changes in the particle's behavior. This can be useful for understanding how a particle responds to different conditions or for predicting future behavior.

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