Applications of derivatives in motion physics

Question

I am a first year engineering student, currently having difficulty understanding the practical application of derivatives in motion problems.

Specifically, I am working on a problem related to uniformly varying rectilinear motion, which requires determining the time at which an object's velocity reaches its maximum value, based on the position equation s(t) = -2t³ + 9t² + 12t + 5.

I understand that I need to use derivatives to find velocity, but am not sure how to approach finding the extreme values ​​of the velocity function.

I hope teachers or friends can explain step by step to help me understand better!

Answer 1

assuming t≥0

s(t) = -2t³ + 9t² + 12t + 5 is the position on a number line at time t

if you want to know extreme of position values, set ds/dt=0,,,ds/dt=s'

at times where ds/dt=0 use those times to find out s(t)

and check boundary condition(s) t=0, s(t)=5, as t->infinity s(t)->-infinity

To find when velocity reaches extreme values set d²s/dt²=0,,,d²s/dt²=s''

s'=-6t²+18t+12

s''=-12t+18

see that s''=0 when t=3/2 units of time

at t=3/2 s'=ds/dt=-6(9/4)+18(3/2)+12=(156-54)/4=51/2 unit distance/unit time

at t=0 (boundary condition) s'=ds/dt=12 unit distance/unit time

51/2>12 so object reaches its maximum velocity at t=3/2 time unit

note: as t->infinity the velocity s'->-infinity which says there is no minimum velocity.

Answer 2

The position function reaches its max or min values when the velocity equals 0 (or at the endpoints of the time interval).

s'(t) = -6t²+18t+12

Set that equal to zero and solve for t using the quadratic formula or completing the square.

Completing the square:

-6t²+18t+12=0

t²-3t-2=0

t²-3t = 2

t²-3t + 9/4 = 17/4

(t-3/2)²= 17/4

t - 3/2 = ±√17/2

t = 3/2 ± √17/2

t₁ = (3 + √17)/2

t₂=(3 - √17)/2

Try visiting this Desmos graph to get a better picture:

desmos.com/calculator/exgjue6vhc

Whoops! I did not read the post carefully enough. The ask is when does the velocity reach its maximum value. That happens when the 2nd derivative is equal to zero. Here is an updated graph:

desmos.com/calculator/80jeqwdjur