Find time t at which the speed of the particle is minimized

In summary, the conversation discusses finding the time at which the speed of a particle, given by the position function r(t), is minimized. The suggested method is to find the derivative of the velocity function and solve for the time at which it equals zero. The concept of finding the absolute value of a vector and minimizing it is also discussed. It is noted that this is the same as minimizing the dot product of the vector with itself. It is then explained how this relates to the Pythagorean theorem in three dimensions. Finally, it is concluded that all mathematics is "common sense once you think about it the right way".
  • #1
popo902
60
0

Homework Statement



If r(t) = −3t2i+2tj+(t2 −3t)k gives the position
of a particle at time t, find the time t at which the speed of the
particle is minimized.


Homework Equations





The Attempt at a Solution



i know for minimization and maximization, you find the derivative and solve for the variable and see which values produces a largest or smallest value when plugged into the original equation right?

so, since this is asking for speed, would it be right to find the deriv of this (v(t)) and find the absolute value of it (|(v(t)| = s(t)) then find the derivative of s(t) and solve for t?
and just a side question but, s(t) is just the magnitude of the velocity vector right?
 
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  • #2
How can you take the absolute value of a vector? Do you mean that you want to dot it with itself? That would give you a scalar valued function of t that would have the value of the length of the velocity vector squared at a particular t.

You could then minimize that.
 
  • #3
Yes, I think that's all pretty right.
 
  • #4
so basically minimize s(t) not s'(t)?
 
  • #5
Well, yeah. Minimize s(t)=|v(t)|. Which as aPhilosopher pointed out is the same as sqrt(v(t).v(t)). Which in turn is minimized at the same time as v(t).v(t).
 
Last edited:
  • #6
i didn't know that s(t) was the same as sqrt( v(t) dot v(t) ), is this a theorem...or just common sense? X|
 
  • #7
What's your expression for the magnitude of a vector v(t)? Isn't it the same as sqrt(v(t) dot v(t))?
 
  • #8
It's sooooort of common sense once you've been shown how to think about it the right way.

In one definition, you have the magnitude of the vector squared times the cosine of zero.

In the other definition, you have (x1, x2, x3)2 = x12 + x22 + x32. This is just one side of the Pythagorean theorem in three dimensions. The other side is the length squared.
 
  • #9
aPhilosopher said:
It's sooooort of common sense once you've been shown how to think about it the right way.

In one definition, you have the magnitude of the vector squared times the cosine of zero.

In the other definition, you have (x1, x2, x3)2 = x12 + x22 + x32. This is just one side of the Pythagorean theorem in three dimensions. The other side is the length squared.
All mathematics is "common sense once you think about it the right way"!
 
  • #10
HallsofIvy said:
All mathematics is "common sense once you think about it the right way"!

Ahhhhhh, but is thinking about it the right way "common sense"? Perhaps you hold common sense in higher esteem than I do ;)
 
  • #11
Look carefully at what I said, "Philospher"! I did not say that "looking at it the right way" is common sense. I said that, once you have looked at it the right way, the rest is "common sense".
 
  • #12
If by common sense, you mean easy, or at least doable, then I agree whole heartedly! I was just taking a dig at common sense. My apologies for the confusion.
 

1. How do you find the time at which the speed of a particle is minimized?

The time at which the speed of a particle is minimized can be found using the derivative of the speed function. This is because the minimum speed occurs at the point where the slope of the speed function is equal to zero. By setting the derivative equal to zero and solving for time, you can determine the time at which the particle's speed is minimized.

2. What is the significance of finding the time at which the speed of a particle is minimized?

Finding the time at which the speed of a particle is minimized is important because it allows us to determine the point at which the particle is moving the slowest. This can be useful in various applications, such as optimizing the efficiency of a machine or predicting the trajectory of a moving object.

3. Can the time at which the speed of a particle is minimized be negative?

Yes, the time at which the speed of a particle is minimized can be negative. This typically occurs when the particle is moving in a backwards direction, such as when it is slowing down before changing direction. In this case, the negative time represents the time before the particle reaches its minimum speed.

4. Are there any other methods for finding the time at which the speed of a particle is minimized?

Yes, there are other methods for finding the time at which the speed of a particle is minimized. One alternative method is to graph the speed function and visually determine the point at which the slope is zero, indicating the minimum speed. Another method is to use calculus techniques, such as the second derivative test, to confirm the existence of a minimum speed and determine the corresponding time.

5. How can the time at which the speed of a particle is minimized be applied in real-world scenarios?

The time at which the speed of a particle is minimized can be applied in various real-world scenarios. For example, it can be used in engineering to optimize the speed of a machine or in physics to predict the motion of an object. It can also be applied in sports, such as determining the best time for a runner to take a rest during a race to minimize their overall speed.

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