The two equivalent parallel velocity vectors

In summary: So parallelness of velocity vectors is given, but parallelness of acceleration vectors is not necessarily given.In summary, the conversation discusses proving that at each point of a curve C, the velocity vectors associated with two equivalent functions X and Y are parallel, but the corresponding acceleration vectors may not be. The method for computing velocity and acceleration vectors at a certain point is also discussed.
  • #1
Dinheiro
56
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This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it:
Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)].
Prove that at each point of C the velocity vectors associated with X and Y are parallel, but
that the corresponding acceleration vectors need not be parallel.
I would really appreaciate some enlightenment.
 
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  • #2
Dinheiro said:
This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it:
Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)].
Prove that at each point of C the velocity vectors associated with X and Y are parallel, but
that the corresponding acceleration vectors need not be parallel.
I would really appreaciate some enlightenment.
How are velocity and acceleration vectors defined at a certain point, i.e. how would you compute them?
 
  • #3
Here's what I tried:
Let r(t) be the position vector and T(t) such that:
T(t) = r'(t)/||r'(t)||
Now, let w(t) be the speed function and v(t) the velocity vector such that:
v(t) = (w(t)*T(t))
Deriving:
v'(t) = w'(t)T(t) + w(t)T'(t) = a(t)
So acceleration is written by the sum of the two components above. if the velocity of Y(t) = X[u(t)], then we can define Y'(t) = X'(u(t))u'(t) = w(t). Substituting this relation in the acceleration equation above, and we get that their acceleration can be defined as:
[X''(u(t))u'(t) + X'(u(t))u''(t)]T(t) + X'(u(t))u'(t)T'(t) = a(t)
Y''(t)T(t) + Y'(t)T'(t) = a(t)
... How should I've written?
 
  • #4
This appears to be a little over complicated to me. The more since you have to deal with the crucial part anyway.
The little given by the text, I assume that ##X## and ##Y## are parametrizations of ##C##. That are functions ##X\, ,\, Y : \mathbb{R} \rightarrow C##. Wouldn't velocity simply be the change of position related to the change in time at a certain point in time?
But this sounds to me like ## \frac{d}{dt} X(t) |_{t=t_0}## and likewise for ##X(u(t))##. The acceleration then is the second derivative.
Note that in the text "at each point" is given which means the differentials have to be evaluated at this point.
 

FAQ: The two equivalent parallel velocity vectors

1. What are the two equivalent parallel velocity vectors?

The two equivalent parallel velocity vectors refer to two vectors that have the same magnitude and direction, and therefore, represent the same velocity.

2. How do you calculate the equivalent parallel velocity vectors?

The equivalent parallel velocity vectors can be calculated using vector addition. This involves adding the x-components and y-components of the two vectors to find the resultant vector, which represents the equivalent parallel velocity.

3. Can the equivalent parallel velocity vectors be negative?

Yes, the equivalent parallel velocity vectors can be negative. This indicates that the velocity is moving in the opposite direction of the chosen coordinate system. However, the magnitude of the vectors will still be equivalent.

4. Why are equivalent parallel velocity vectors important in physics?

Equivalent parallel velocity vectors are important in physics because they allow us to simplify complex motion problems and analyze the motion of an object in a more straightforward manner. They also provide a more accurate representation of an object's motion in a particular direction.

5. How are equivalent parallel velocity vectors used in real-world applications?

Equivalent parallel velocity vectors are used in various real-world applications, such as calculating the motion of objects in sports, predicting the path of projectiles in physics, and analyzing the motion of vehicles in engineering. They are also essential in understanding the motion of fluids in fluid mechanics and weather patterns in meteorology.

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