Hello, When doing problems with SHM, my textbook says something like: An object in vertical shm is described by <insert some function>. Find the speed after X seconds. my question is, how do you know if the function is referring to the position of the object, or the velocity, or accelerration? What are some key words to look for? Thanks.
"something like" is not usually helpful - how are we to tell between your interpretation of what it is like and what it says? The function itself will use variable names that are suggestive of what it is in terms of ... i.e. an "x" would be positions and a "v" would be velocity. They may use dots to refer to time derivatives in the normal way. An equation may be in terms of several ... eg, the acceleration for a mass-on-spring typically depends on position as ma=-kx, where m and k are constants. What you are seeing is probably the result: y(t)=Asin(wt+d) or some variation right? So this is a position equation.... A w and d are constants for the system. v(t)=Bsin(ut+e) would be a velocity equation, B u and e are constants. If it is the same system, the constants will be related to each other.
No, one can use either sine or cosine for the position function, which gives correspondingly cosine or sine for the velocity (and negative sine or negative cosine for the acceleration). Some books do it one way, some books do it the other way.
For a mass on a spring: x(t)=Asin(wt) would be the case where you started your stopwatch (t=0) when the mass passed through it's equilibrium position and headed in the +x direction and v(t)=wAcos(wt)=wAsin(wt+π/2) But it's your stopwatch: you can start it whenever you like. If you had waited to start your stopwatch when it was going through the equilibium position the other way, then it would be x(t)=-Asin(wt) If you started the stopwatch from when the mass has it's maximum displacement in the +x direction, then it would be x(t)=Acos(wt). But if you just started at any old place it would be: x(t)=Asin(wt+a)=Acos(xt+b) ... either sine or cosine could be used but b and a will be different. Can you see what the difference |a-b| has to be? Without the x(t)= part at the start, there would be no way to tell if the Asin(wt) refers to displacement rather than acceleration or velocity.
its giving you the position with time, as simple as that!!!!, to find the speed, just differentiate the function and plug in the value of time
Yep, a question worded like that almost always gives the position (or displacement) as a function of time. When it doesn;t, the nature of the function should be clear from the rest of the text, the variables used, and/or by applying your knowledge of SHM ;)