Minimum Value of Particle in Space

[mill]

Homework Statement

At what time t does the speed of the particle moving in space with its position function r(t)=##<t^2, 3t, t^2 - 8t>## have its minimum value?

Homework Equations

Derivative, speed

The Attempt at a Solution

Found derivative.
r'=<2t, 3, 2t-8>
Found speed.
|r'|=##\sqrt {4t^2 + 9 + (2t -8)^2}##
simplified

set speed to 0

##4t^2 + 9 = -(2t-8)^2##
##9=-(t^4 - 12t^2 +64)##
##-55=-t^2 (t^2 +12)##

The answers I'm getting from this setup are really off. I'm not sure where I am going wrong. t=sqrt(55) or sqrt(67)...but the answer is t=2.

 

[CAF123]

|r'|=√(4t² + 9 + (t² -8)²)

You have the 2 in the wrong place there.

set speed to 0

You do not want to set the speed to zero. What should you be setting to zero?

 

[mill]

You have the 2 in the wrong place there.



You do not want to set the speed to zero. What should you be setting to zero?

Fixed the two typo. Everything following is the same though.

I set the equation to 0 in order to find the min. or is that not what I'm supposed to do?

 

[CAF123]

Fixed the two typo. Everything following is the same though.

It should not be, because you will no longer get a quartic in t.

I set the equation to 0 in order to find the min. or is that not what I'm supposed to do?

You want to find the value of t at which the speed is minimized. You have an expression for the speed of the particle at any time t. What should you do with this expression to obtain its minimum?

 

[mill]

It should not be, because you will no longer get a quartic in t.You want to find the value of t at which the speed is minimized. You have an expression for the speed of the particle at any time t. What should you do with this expression to obtain its minimum?

It's not covered by example in my books, so I was confused about the setup. While looking around, I've seen |r''| and |r'|^2 being thrown around in these types of problems but I'm not sure how they're being used. Ideally to minimize, I would set the chosen equation to 0 to find the critical points. If I can't set |v| to 0 what do I set it to?

 

[CAF123]

What you have found is $$\left|\frac{\text{d}\vec r}{\text{d} t}\right| = |\vec v | = f(t).$$ By setting this to zero, you are minimizing r not v. So to minimize v, you should...?

 

[Ray Vickson]

It's not covered by example in my books, so I was confused about the setup. While looking around, I've seen |r''| and |r'|^2 being thrown around in these types of problems but I'm not sure how they're being used. Ideally to minimize, I would set the chosen equation to 0 to find the critical points. If I can't set |v| to 0 what do I set it to?

So, you have not covered problems of maximizing or minimizing functions such as ##s(t) \equiv |r'(t)|?## Forget about ##r''## or whatever; just look at the function ##s(t)##---you have a formula for it in terms of ##t##, and that's all you need.

 

[SammyS]

It's not covered by example in my books, so I was confused about the setup. While looking around, I've seen |r''| and |r'|^2 being thrown around in these types of problems but I'm not sure how they're being used. Ideally to minimize, I would set the chosen equation to 0 to find the critical points. If I can't set |v| to 0 what do I set it to?

What do you do to find the minimum or maximum of any function in general, in this case a function for which time, t, is the independent variable?