Question

A particle moves alone a line so that its position at any time t>= 0 is given by the function
s(t) = 4t^3 -2t^2 +1

a) find the displacement during the first 3 seconds.
b) Find the average velocity during the first 3 seconds.
c) Find the instantaneous velocity during the first 3 seconds.
d) Find the acceleration of the particle when t = 3
e) At what value of values of t does the particle change direction

Answer 1

for a, i got 91, and for d i got 68, am i correct? and how do i do the rest?

for b and c, im confused between the difference between finding instant and average

Answer 2

i found the s'(t) =12t^2-4t
and s''(t) to be 24t - 4

Answer 3

yes thats correct

Answer 4

well doesn't the particle start, t = 0, 1 unit right of the origin. which affects the displacement..?

Answer 5

how is t=0 a unit right of the origin?

Answer 6

for b find f(0) and f(3)

then its \[avg vel = \frac{f(3) - f(0)}{3 - 0}\]

Answer 7

o wait, i see

Answer 8

substitute t = 0 into the displacement equation, what value do you get...?

Answer 9

so a is 90, rather than 91,

Answer 10

that's correct.

(c) instantaneous velocity, find the s'(t) then substitute t = 3

Answer 11

oops average velocity should be s(0) and s(3)

Answer 12

average velocity does not use s', but instead uses s?

Answer 13

for the average velocity use\[avg..vel = \frac{s(3) - s(0)}{3 - 0}\]

Answer 14

so u dont use the derivative of the displacement formula to find the average velocity?

Answer 15

for average velocity use the displacement equation. All you are doing is finding 2 ordered pairs (0, s(0)) and (3, s(3)) and then finding the slope of the segment. Which is average velocity

Answer 16

ah, i see, but how do i do e, what am i supposed to find?

Answer 17

find the stationary point(s) by solving the 1st derivative... this is where velocity is zero and an the motion changes...

Answer 18

ah, so we need to find the horizontal tangent line?

Answer 19

no, you just need to find the values of t, where the motion changes...

Answer 20

ya, where s'(t) = 0

Answer 21

but solving the 1st derivative is part way to finding the equation of the tangent. But not needed... just the values of t

Answer 22

that's correct and then just solve for t.

Answer 23

tyvm