Instantaneous rate of change

[fitz_calc]

Homework Statement

s is the position of a particle along the s-axis, find an expression for the velocity and acceleration and determine when v=0

s=1-3t^2

Homework Equations

v=ds/dt , a=dv/dt

The Attempt at a Solution

No idea where to begin, my book is not very clear. Do I just set v=0 equal to s = 1-3t^2? I know the answer is v=-6t but do not know where to begin - thanks!

 

[berkeman]

Yes, differentiate s(t) as you have shown to find v(t). Then set v(t) = 0, and solve for t. You're on the right track.

 

[fitz_calc]

I still do not know where to begin. my book gives an example s=2t^2-4t. when they begin to solve the problem they go from v=ds/dt to v=4t-4

i don't see how they come to this value?

 

[PowerIso]

They found the derivative of 2t^2 -4t.

 

[fitz_calc]

so i found acceleration to be -6 by taking the derivative of -6t. what does this mean in layman's terms, exactly? I don't understand how taking the derivative of s yields the velocity, and taking the derivative of velocity yields acceleration...

 

[berkeman]

so i found acceleration to be -6 by taking the derivative of -6t. what does this mean in layman's terms, exactly? I don't understand how taking the derivative of s yields the velocity, and taking the derivative of velocity yields acceleration...

Velocity is the change in position per unit time, or ds/dt. Acceleration is the change in velocity per unit time, or dv/dt.

It's helpful to carry along your units...

** units of position s are meters [m]

** units of velocity v are meters per second [m/s]

** units of acceleration a are meters per second squared [m/s^2]

When you get an answer like a = -6 [m/s^2], that means that you have a *deceleration* of 6 [m/s^2]. Does that help?

 

[HallsofIvy]

so i found acceleration to be -6 by taking the derivative of -6t. what does this mean in layman's terms, exactly? I don't understand how taking the derivative of s yields the velocity, and taking the derivative of velocity yields acceleration...

It means that this object has a constant downward acceleration, just a something falling in constant gravity does.

Surely you have learned that the derivative is the rate of change of a function relative to the variable? If x(t) is distance, x, as a function of time, t, then dx/dt is the rate of change of distance relative to time: precisely the instantaneous velocity. Of course, the second derivative then is the rate of change of velocity relative to time: the definition of acceleration.

 

[fitz_calc]

i get it now, some of these topics were covered in my mechanics course last year, though we never really touched on derivatives like this technical calculus course is. thanks.

 

[HallsofIvy]

Oh, God, I hate courses like that. I once taught a "Calculus for Economics and Business Administration" course. The textbook I was required to use covered limits in one page just listing the three limit laws:
If $\lim_{x\rightarrow a} f(x)= L_1$ and $\lim_{x\rightarrow a} g(x)= L_2$ then $\lim_{x\rightarrow a} f(x)+ g(x)= L_1+ L_2$.
If $\lim_{x\rightarrow a} f(x)= L_1$ and $\lim_{x\rightarrow a} g(x)= L_2$ then $\lim_{x\rightarrow a} f(x)g(x)= L_1L_2$
If $\lim_{x\rightarrow a} f(x)= L_1$ and $\lim_{x\rightarrow a} g(x)= L_2$ AND $L_1$ is not 0, then $\lim_{x\rightarrow a} f(x)/g(x)= L_1/L_2$.

On the very next page, they defined the derivative as
$$\frac{df}{dx}(a)= \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$$
not bothering to point out that the laws of limits they had given do not apply here because the denominator necessarily does go to 0!

They had left out the most important law of limits:
If f(x)= g(x) for all x except x=a, then $\lim_{x\rightarrow a} f(x)= \lim_{x\rightarrow a}g(x)$!