Analyzing Bee's Spiral Path: Velocity & Acceleration

  • Thread starter chaotixmonjuish
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In summary, the conversation discussed the path of a bee in a spiral motion and how the angle between its velocity and acceleration vectors remains constant. The solution to finding the dot product was also provided, resulting in a final answer of cos-1(a bunch of constants) in the range of 0 and pi.
  • #1
chaotixmonjuish
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1.)

A bee goes from its hive in a spiral path given in plane polar coordinates by
r = b*ekt , θ = ct,
where b, k, c are positive constants. Show that the angle between the velocity vector and the
acceleration vector remains constant as the bee moves outward. (Hint: Find v · a/va.)

so here is my v and a

2.)
v = (r')er+(r*θ')eθ

a = (r''+rθ')er+(rθ''+2r'θ')eθ

r' = bk*ekt
r'' = bk2ekt
θ' = c

3.) my attempt at a solution

bkekt(bk2ekt-bektc2)+(bektc)(2bkektc)

is that the right dot product

this is where I'm stuck
 
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  • #2
so I got down to this:

e2kt(a bunch of constants)+e2kt(a bunch of constants)/e4kt


all the e's canceled out and left just constants
 
  • #3
My answer is for the dot product is:

e2kt(b2k3 + b2kc + 2b2kc2)
 
  • #4
you continue by finding the modulus of v and a then by using the hint you cross out the

b2e2kt and end up with cos-1(a bunch of constants) and the answer
 
  • #5
.

I would first clarify the question and make sure I understand what is being asked. From the given information, it seems like we are trying to prove that the angle between the velocity vector and the acceleration vector remains constant as the bee moves outward in its spiral path.

To do this, we can use the dot product formula:

v · a = |v| |a| cosθ

where θ is the angle between the two vectors.

From the given equations, we can calculate the magnitude of the velocity vector as:

|v| = √(v · v) = √[(bk*ekt)^2 + (bektc)^2] = bk*ekt

Similarly, the magnitude of the acceleration vector is:

|a| = √(a · a) = √[(bk2ekt + bektc^2)^2 + (2bkektc)^2] = √(bk2ekt + bektc^2)^2 + 4b2k2e2ktc^2] = bk2ekt + 2bkektc

Now, to find the angle θ between the two vectors, we can use the dot product formula again:

cosθ = (v · a)/(|v| |a|)

Substituting the values we calculated above:

cosθ = [(bkekt)(bk2ekt + 2bkektc)]/(bk*ekt)(bk2ekt + 2bkektc) = 1

This shows that the angle between the velocity vector and the acceleration vector is always cos^-1(1) = 0, which means it remains constant as the bee moves outward in its spiral path.

Therefore, we have proven that the angle between the velocity vector and the acceleration vector remains constant.
 

1. What is the purpose of analyzing a bee's spiral path?

The purpose of analyzing a bee's spiral path is to gain a better understanding of the bee's flight patterns and behavior. This information can be used to study the effects of environmental factors, such as wind and temperature, on bee flight and to improve agricultural practices for pollination.

2. How is the velocity of a bee's spiral path calculated?

The velocity of a bee's spiral path can be calculated by dividing the distance traveled by the bee by the time it took to travel that distance. This can be done by tracking the bee's position at regular intervals and using the formula: velocity = distance / time.

3. What factors can affect a bee's velocity and acceleration?

Several factors can affect a bee's velocity and acceleration, such as wind speed and direction, temperature, humidity, and the bee's age and health. These factors can impact the bee's flight efficiency and its ability to navigate and collect nectar.

4. How does analyzing a bee's spiral path help with bee conservation?

Studying a bee's spiral path can provide valuable information for bee conservation efforts. By understanding the bee's flight patterns and behavior, we can identify potential threats to their survival, such as changes in air quality or habitat loss. This information can then be used to develop strategies for protecting and preserving bee populations.

5. What are some potential applications of analyzing bee flight patterns?

Analyzing bee flight patterns can have various applications, such as improving agricultural practices for pollination, developing more efficient and sustainable methods of beekeeping, and studying the impact of environmental factors on bee populations. It can also provide insights into the flight patterns of other pollinators and how they contribute to ecosystem health.

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