Physics -trying to prove something?

Question

Given that x=Acos(wt) is a sinusoidal function of time, show that v(velocity) and a ( acceleration) are also sinusoidal functions of time.

Answer 1

You are given the position (x) of an object as a function of time (t).

x = A * cos (ωt)

The first derivative of position with respect to time is the velocity.
dx/dt = v

Taking the derivative of the position function we are given, we get,
v = -ω * A * sin (ωt)
Since A is just a constant and the derivative of cosine is negative sine multiplied by the derivative of the function of t inside the cosine function (the chain rule).
We now know the velocity of the object as a function of time.

The second derivative of position with respect to time (which is also the first derivative of velocity with respect to time) is acceleration.
d^2x/dt^2 = dv/dt = a

Taking the derivative of the velocity function we just found we get,
a = -ω^2 * A * cos (ωt)
Since A and ω are just constant and the derivative of a sine function is cosine multiplied by the derivative of the function of t inside the sine function (chain rule again).

We now know equations for the position, velocity, and acceleration of the object as a function of time.
And what do we notice about these equations….?
They are all sinusoidal (or co-sinusoidal, which is just phase shifted from sine) functions of time.
We could go on and on taking derivatives, but we are always going to keep getting sine and cosine functions for our result since the derivative of sine is cosine and the derivative of cosine is negative sine.


EDIT:
In general, when a question asks you to “prove” something, it is not a good idea to use the word, “obvious” in your answer, it somewhat defeats the purpose of a proof. They are asking you to “show” something, so show it.

Answer 2

You need to know some calculus to answer this question. I hope you do. If that's the case, all you need to do is take the derivative of both sides of the position equation - the derivative of position is velocity, and the derivative of Acos(wt) will be some variant of the sine function - because you should know if you've taken any useful calculus that derivatives of sinusoidal functions always bring you back to other sinusoidal functions.

If you don't know calculus, this is a really hard question and they shouldn't really be asking you about it.

Answer 3

This is OBVIOUS !

v = dx/dt = - w A sin wt

a = dv/dt = d^2 x/dt^2 = - w^2 A cos wt

Both sin wt and cos wt are "sinusoidal functions of time."

Live long and prosper.

Answer 4

Differentiate twice: x' = - Aw sin(wt); x" = -Aw*2 cos (wt). Which are obviously sinusoidal.