Find the acceleration?

Question

At time t, the position of a body moving along the s-axis is  s = t³ -27t² +240t m. Find the body's acceleration each time the velocity is zero.

Answer 1

The given position function:

s(t) = t³ -27t² +240t

The velocity function (derivative of the position function):

v(t) = d/dt[s(t)] = 3t² - 54t +240

Finding times where velocity us 0, or v(t) = 0:

v(t) = 0

3t² - 54t +240 = 0 divide both sides by 3

t² - 18t + 80 = 0 factor

(t - 10)(t - 8) = 0

So velocity is 0 at t = 10 and t = 8. Plug these into the acceleration equation (which is the derivative of the velocity equation):

a(t) = d/dt[ v(t) ] = 6t - 54

a(10) = 6(10) - 54 = 60 - 54 = 6

a(8) = 6(8) - 54 = 48 - 54 = -6

So the body's acceleration is -6 when velocity is zero the first time, and 6 when velocity is 0 the second time. (Let me know if there are any errors or if anything is unclear.)