Suppose I was asked to find the Maximum acceleration of particle given the equation:
s(t) = -16t^2+7t+2
I know that
Velocity:
v(t) = -32t+7
and that
Acceleration
a(t) = -32
But how would I Maximize it? I know that when it comes to velocity I would just take the derivative of the given s(t) equation, then set that velocity v(t) equation equal to zero and solve for t. Then I would plug in that t value in the the original s(t) height equation; however I am unsure as to how I would go about doing this for acceleration. I know that F'' aka the acceleration double derivative deals with concavity. Would the best method be to set up a test line then test values below -32 and above -32 to see whether it is concave up or concave down. I guess that I would also have to test -32 to see whether or not it would be equal to. I am just wondering how I would go about "Maximizing Acceleration". Please let me know when you all get a chance. As always I appreciate all of your help and insight.
remember, acceleration is different from velocity. velocity is the rate of change of speed in a particular direction, acceleration is the rate of change of velocity.
to find the max of any function, you would do as you say, find its derivative, set it to zero, and examine the function at that point. if there is no variable in the derivative of a function, it means it is constant, so everything stays the same, and the max (and min for that matter) is just the value of the function
Right... as Jhevon said, to maximize or minimize any function, take its derivative, find its critical values and examine the its increasing/decreasing behavior. You obviously know how to maximize velocity since you explained the process in the original post. To maximize acceleration, take the third derivative of position (or first derivative of F''), which I think is known as 'jerk' (but don't quote me on that), set it equal to zero, and use the same process. There is a slight difference though: F'' = 32 so F''' = 0 (i.e. the object is neither accelerating or decelerating). The max acceleration is therefore 32 m/s^2.
Given a displacement vector s, the velocity vector v, and acceleration vector a, we define a vector j:
j = da/dt = d^2v/dt^2 = d^3s/dt^3
as the "instantaneous jerk" on an object. Jerk doesn't stand for anything but how hard you might "yank" on a rope to make something move. (I love some Physics terminologies!)
Naturally then, the vector (Delta)a/(Delta t) is me, your "average jerk."
-Dan