What is the Time When the Particle Has Zero Acceleration?

[Destrio]

A particle moves along the x-axis, its position at time t is given by
x(t)= 5t/(10+7t^2)

where t is >= 0 and measured in seconds and x is in meters

The acceleration of the particle equals 0 at time t = ?

I took the 1st derivative and got
i got v(t ) = 5(10-7t^2) / (10+7t^2)^2

or i got v(t ) = 5(10-7t^2) * (10+7t^2)^-2

then i took the derivative of that and got:

a(t ) = 5[(10-7t^2)(-2)(10+7t^2)^(-3) *(14t) + (10+7t^2)^2 *(10-7t^2)]

so i tried factoring out (10-7t^2), to get one answer which i got as sqrt(10/7)

and then for the other answer i got:
(10+7t^2)^5 = 28t

which i simplified to:
2401t^8 + 13720t^6 + 29400t^4 + 28000t^2 - 28t + 10000 = 0

now I'm not sure if I'm on the right track, and if I am, I'm not sure how I can solve for t.

Any help is much appreciated, I've been working on this (my first problem) for about an hour and I have a midterm in a few days :s

thanks

 

[Galileo]

v(t ) = 5(10-7t^2) * (10+7t^2)^-2

then i took the derivative of that and got:

a(t ) = 5[(10-7t^2)(-2)(10+7t^2)^(-3) *(14t) + (10+7t^2)^2 *(10-7t^2)]

Just keep the powers of (10+7t^2)^2 in the denominator like you did when calculating v. The numerator is then easy to factorize.

 

[Destrio]

the numerator will still end up being 2401t^8 + 13720t^6 + 29400t^4 + 28000t^2 - 28t + 10000
if i make a common denomintor with the two terms
ahhh

 

[Galileo]

Don't write out the brackets, that's going to be horrible. If you never write out the brackets in your calculations you'll end up with a form that's already quite nicely factorized.

If you apply to quotient rule to:
5(10-7t^2)/(10+7t^2)^2
without expanding, you get (10+7t^2)^4 in the denominator right? YOu'r numerator will not be too bad either. Just look for common terms.

 

[Destrio]

what do you mean don't write out the brackets?

applying the quotient rule i got:
5[(10+7t^2)^2 *(-14t) - (10-7t^2)(2)(10+7t^2)(14t)]/(10+7t^2)^4

so I can factor out 14t
and 10+7t^2

will my two terms be
10+7t^2
and
10-7t^2
?

thanks

 

[Destrio]

why did I get something different from when I used to product rule?
or were they equivalent but arranged differently?

 

[Galileo]

You made a small error when using the product rule. The last answer you gave using the qoutient rule is correct and it should be easy finding the zero's now.