Solving a 3 variables equation, one happens to cause an oddity

Jun 2018
24
0
Jordan
Hey, so we had this question in class:






Given that the relationship between distance (m) and velocity (v) of an object is \(\displaystyle v^2 = 1 - m^3\)




Find the acceleration of the object when \(\displaystyle m=1\)




By taking the derivative of each side with respect to \(\displaystyle t\)




\(\displaystyle 2v \frac{dv}{dt} = -3m^2 \frac{dm}{dt} \)




and we know that\(\displaystyle \frac{dv}{dt}\) = acceleration, \(\displaystyle \frac{dm}{dt} = v\), then:




\(\displaystyle 2v a = -3m^2 v\)




and by solving for \(\displaystyle a\) without dividing by \(\displaystyle v\) since \(\displaystyle v \) can be zero(assuming that we don't know yet, \(\displaystyle a = -3/2\)




But hold on... we've just substituted m with 1, this means that we have to substitute v with zero, but if we do this we would turn out with 0 = 0, without solving for a




If we don't substitute v with 1, we wouldn't be able to solve for acceleration if we cancel v's with dividing, and it doesn't really make sense to keep v in there while we know that v = 0 when m = 1. My teacher clarified that by going back to the definition of derivative, that it's after all nothing but a limit and a limit is an approximation of some f(x) when x approaches some value, so a limit which equals 0 would be in real something like 0.0001 or 0.000000001 in a way we would be able to cancel v with v. But that doesn't really make sense to me. If that's true, when to use the approximate definition of limits and when to use the accurate? And what can we do for this equation?
 
Dec 2015
1,084
169
Earth
If not divide by \(\displaystyle v\)
\(\displaystyle v(2a+3m^2)=0\) or \(\displaystyle 2a+3m^2=0\)
\(\displaystyle a=\frac{-3m^2 }{2}\)
 
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topsquark

Math Team
May 2013
2,532
1,051
The Astral plane
Meaningless question since the units don't match.
Clearly if we were to be more critical about the relationship \(\displaystyle v^2 = 1 - m^3\) then we should be writing \(\displaystyle v^2 = p - qm^3\) where p and q are constant with the appropriate units. The we just say that |p| = 1 and |q| = 1.

I've seen this kind of thing a lot when going from a Physics problem to a Calculus problem.

-Dan
 
Jun 2018
24
0
Jordan
Treat it as an impartial differentiation problem along with chain rule, and taking derivatives with respect to time for now.


If not divide by \(\displaystyle v\)
\(\displaystyle v(2a+3m^2)=0\) or \(\displaystyle 2a+3m^2=0\)
\(\displaystyle a=\frac{-3m^2 }{2}\)
Yes I did mention this implicitly in my post, but why can't we substitute for \(\displaystyle m\) directly in the equation rather than simplifying it to this form?

Also, is what my teacher said true?
 
Dec 2015
1,084
169
Earth
Let's take a simple example with velocity.
If \(\displaystyle s=s(t)\) then \(\displaystyle vt\neq s(t)\)
Once distance is given as function of time then \(\displaystyle v=s’(t)\)
 
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