What Is the Maximum Speed of the Particle and When Is It Reached?

[Easy_as_Pi]

Hey all! Long time lurker, first time poster. Already answered questions on this forum have managed to help me out immensely in calc 1 and 2, but I am truly stumped on this problem o.0 Thanks in advance for your time!

Homework Statement

The position of a particle in the plane at time t is
r(t) = t/sqrt(1+t^2)i + 1/sqrt(1+t^2)j + tk
What is the particle's highest speed. When and where does it reach this speed?

Homework Equations

Speed = magnitude of velocity vector
velocity = derivative of position vector
acceleration = derivative of velocity vector

The Attempt at a Solution

Well, I took the derivative of r(t), and got velocity equal to the following vector
<1/(t^2+1)^3/2 i, -t/(1+t^2)^3/2 j, 1k>
I then took the derivative of velocity to get acceleration, which I found to be:
a(t) = <-3t/(t^2+1)^5/2 i, (2t^2-1)/(t^2+1)^5/2 j, 0 k>

I know I need the zeroes of this, because they will be the maximum / minimum values for my velocity. So, I set each part (i,j,k) equal to 0 and solved. I don't think this is right, however. My answers were 0 and 1/4 for t. With 0 being my minimum and 1/4 being the max. If I plug 1/4 in for t in the v(t) equation and find the magnitude I get speed being equal to sqrt(1273)/27. I'm not sure about the where and when? Would that just be the velocity vector at time t (vector gives where, t gives when)?

Sorry for the lack of latex, I tried to use it, but couldn't get my work to format right.

 

[Dick]

Welcome to the forums as a first time poster. It's not necessarily true that the maximum of speed happens at a point where the acceleration is zero. Besides, your acceleration vector is never zero. Forget the acceleration and just get an expression for speed and maximize it.

 

[Easy_as_Pi]

Hey Dick, thanks for the quick reply. What do you mean get an expression for the speed and maximize it? I'm sorry if some of these questions are silly, but I'm taking this class in summer session, and just started a few days ago. That, coupled with barely being able to understand my professor is making this course a bit of an uphill battle.
I know speed is the magnitude of velocity vector. So, sqrt(1/(t^2+1)^3 + t^2/(1+t^2)^3 + 1). Then take the derivative of this and find it's zeroes? I found a thread where you explained how to find the minimum and am trying to base my answer off that. When I take the derivative of the velocity magnitude I get: -4t/(t^2+1)^3. This results in -4t=0...is there something I'm missing here?

 

[Dick]

You've got the right idea. But I don't get the same expression for the derivative of velocity magnitude you do. |r'(t)|=sqrt(1+1/(t^2+1)^2), right? You can find the max of that by inspection. What you did is find the derivative of |r'(t)|^2=1+1/(t^2+1)^2 Which is actually perfectly ok, if you see why. Either way you do it you get an extreme point at t=0.

 

[Easy_as_Pi]

Ah wow, I went through too quickly and made a dumb algebra mistake. I didn't cancel out the t^2+1, and instead went on to make the equation more complicated than it needed to be. I tripped myself up. After reworking, I got the same answer as you. This just confuses me a bit: how can a particle be at it's maximum speed when time is 0?

Thanks again for your help. It's much appreciated. I spent a good while with my classmate trying to figure this one out, and we were stumped.

 

[Dick]

Ah wow, I went through too quickly and made a dumb algebra mistake. I didn't cancel out the t^2+1, and instead went on to make the equation more complicated than it needed to be. I tripped myself up. After reworking, I got the same answer as you. This just confuses me a bit: how can a particle be at it's maximum speed when time is 0?

Thanks again for your help. It's much appreciated. I spent a good while with my classmate trying to figure this one out, and we were stumped.

The speed at t=0 is sqrt(2), right? At t=+/-1 it's sqrt(5/4). That's less. I'm not sure why it should be confusing that you have a max at t=0.

 

[Easy_as_Pi]

Hey again. I'm not having any issues with the computation of it. I see why mathematically t=0 is the max, it's the conceptual part that I'm lost on. I just don't see how a particle could be moving, let alone at it's max speed when time, technically, hasn't started moving yet (t=0)?

 

[Dick]

Hey again. I'm not having any issues with the computation of it. I see why mathematically t=0 is the max, it's the conceptual part that I'm lost on. I just don't see how a particle could be moving, let alone at it's max speed when time, technically, hasn't started moving yet (t=0)?

Why can't it be moving at t=0? t=0 is just a point in time like any other. There's nothing special about t=0.