When is the speed of the particle a minimum?

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MarkSA

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Sep 8, 2007
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Hi,

Q: Position function of a particle is given by r(t) = <t^2,5t,t^2 - 16t>. When is the speed a minimum?
Thinking back to Calc 1 days, which I only vaguely remember, I thought I would need to take the second derivative and find at which values of t it equals zero to get the critical points... because the velocity should be at a minimum when speed is at a minimum since speed = |v|.

r'(t) = <2t,5,2t - 16>
This is my velocity function then.. derivative of that (acceleration):
r''(t) = <2,0,2>

Ok, this is my problem. r''(t) = 2i + 0j + 2k. But it's not a function of time anymore. So it can't ever = 0? I'm guessing I am messing up somewhere. Can you help?
 
Re: When is the speed a minimum?

MarkSA said:
Hi,

Q: Position function of a particle is given by r(t) = <t^2,5t,t^2 - 16t>. When is the speed a minimum?
Thinking back to Calc 1 days, which I only vaguely remember, I thought I would need to take the second derivative and find at which values of t it equals zero to get the critical points... because the velocity should be at a minimum when speed is at a minimum since speed = |v|.

r'(t) = <2t,5,2t - 16>
This is my velocity function then.. derivative of that (acceleration):
r''(t) = <2,0,2>

Ok, this is my problem. r''(t) = 2i + 0j + 2k. But it's not a function of time anymore. So it can't ever = 0? I'm guessing I am messing up somewhere. Can you help?
Click to expand...

Remember speed is the magnitude of the velocity vector.
 
Re: When is the speed a minimum?

as subhotosh said, speed is the magnitude of the velocity vector.

so if r(t) = <t^2, 5t, t^2 - 16t>

you got your derivative right:

r'(t) = <2t, 5, 2t -16>

which is the velocity vector

now, speed is the magnitude of velocity, and taking r'(t) to be a position vector:

speed = |r'(t)| = SQRT( (2t)^2 + (5)^2 + (2t-16)^2 )

simplify this expression, this is your formula for speed.

then examine this function to find its minimum, you will get t = some number

which tells you at what time, t, the speed is at a minimum
 
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