# could be some misprint in my or i the one who dont understand please confirm

if acceleration described as vx=-100x(velocity atvx axis) prove that vx=A sin (10t+Φ) just if the problem says ax = 100x i could say that vx =10x
dv/dt dx/dx= 100x

#### HallsofIvy

"if acceleration described as vx=-100x(velocity atvx axis)" makes no sense. If vx is velocity then that says nothing about acceleration. If ax= -100x (where ax is the acceleration along the x axis) then x= A sin(10t+Φ) since then vx= dx/dt= 10A cos(10t+Φ) and ax= dvx/dt= -100A sin(10t+Φ)= -100x.

Also "dv/dt dx/dx" is just dv/dt since dx/dx= 1. You are probably remembering, incorrectly, dv/dt= (dv/dx)(dx/dt)= v dv/dt.

• 1 person

#### Woody

does it make any more sense if we re-state your main equation as:
vx=-100*(velocity at vx axis)"

I have tried to clarify the use of "x" in the equation.

In one case it is the identifier of v
(ie it is vx as opposed to perhaps vy or vz)

In the other case it is a multiplication symbol, which I have replaced with a *.

first thank you so much for replying and im terribly sorry about my incorrect writing, yes it should ax = -100x could you explain here how it could turn into 10A cos(10t+Φ) sorry im still beginner regarding this matter

"if acceleration described as vx=-100x(velocity atvx axis)" makes no sense. If vx is velocity then that says nothing about acceleration. If ax= -100x (where ax is the acceleration along the x axis) then x= A sin(10t+Φ) since then vx= dx/dt= 10A cos(10t+Φ) and ax= dvx/dt= -100A sin(10t+Φ)= -100x.

Also "dv/dt dx/dx" is just dv/dt since dx/dx= 1. You are probably remembering, incorrectly, dv/dt= (dv/dx)(dx/dt)= v dv/dt.
yes thats really make sense when lined up from x to v then y for now i only understand when ax at written as ax=-100Asin(10t+Φ) but i confuse when its only written as ax=-100x like in my book
and its seems the answer in my books is incorect too since it stated v=Asin (10t+Φ) where this answer should used as x and not v

for your second statement it become like this (please correct me if i wrong im just a noob)
assuming ax=100x (x in meter)
dvx/dt=a
dvx/dt=100x
dxv/dt dx/dx =100x
(dx/dt=vx)
so it became
xv dvx = 100x dx then
∫ xv dvx = ∫ 100x dx
½vx² = ½100x²
in the end vx should have value 10x

#### HallsofIvy

In order to solve this kind of problem you would have to know some basic "differential equations". The equation you have here, ax= dvx/dt= d^2x/dt^2, is a "second order ordinary differential equation with constant coefficients". It can be shown that any solution to such an equation is "of the type" e^{at} for some constant a. The first derivative of that is ae^{at} and the second derivative is a^2e^{at}. Putting those into the equation, ax= d^2x/dt^2= a^2e^{-at}= -100x= -100e^{at}. Dividing both sides of a^2e^{at}= -100e^{at} by e^{at} we have the "characteristic equation" a^2= -100 so a= 10i or a= -10i. That means the general solution is a linear combination of e^{10it} and e^{-10it}.

But since this problem clearly involves only real numbers we would prefer to write the solution using only real numbers. We can do that by remembering that e^{ait}= cos(at)+ isin(at). That is, the solution to a "linear ordinary differential equation with constant coefficients", when the solutions to the characteristic equation are imaginary numbers can be written in terms of sine and cosine.

Of course the problem here did NOT ask you to find the solution- it proposed a solution and asked you to show that it was correct. To do that you only had to use Calculus to show that it did satisfy the conditions.