Question

The position function of a particle in rectilinear motion is given by s(t) = t3 - 12t2 + 45t + 4 for t ≥ 0.Find the position and acceleration of the particle at the instant the when the particle reverses direction. Include units in your answer.

Answer 1

take the derivative, set it equal to zero, and solve

Answer 2

Thank you!

Answer 3

so s'(t)= 3t^2-24t+45 correct?

Answer 4

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Answer 5

so y=0 when x=3 and when x=5, what do we do with this? we need to find the position and the acceleration. does this have anything do do with concavity??

Answer 6

By the way, the acceleration is the second derivative of the position function!
The velocity is just the first derivative of the position function!
Keep that in mind

Answer 7

@sbentleyaz
I know you may be busy, and I am busier, so please respond faster for saving my time and yours!

Answer 8

so we derive it again ?

Answer 9

https://www.desmos.com/calculator/tlb1yxhw7j
Yes, that is right to get the acceleration!

Answer 10

so at x=3 the acceleration is -6 and at x = 5 the acceleration is 6? @3mar

Answer 11

Just derive without substitution with x=3!

Answer 12

what do you mean

Answer 13

what is the function of the velocity?

Answer 14

3x^2-24x+45

Answer 15

Can you derive it now!
Just derive it!

Answer 16

so 6x-24

Answer 17

is that right ?

Answer 18

and are we ignoring x=5?

Answer 19

Just the second derivative represents the accel. without substit.!

Answer 20

but it has to be at the instant the particle reverses direction so why dont we sub in 3

Answer 21

i am very confused

Answer 22

"at the instant the when the particle reverses direction"
Where is that point in your opinion?

Answer 23

when the graph goes negative?

Answer 24

No, when the position changes its direction!

Answer 25

so what does that look like on a graph? is that the minimum

Answer 26

where is the point that has no slope at the curve of the position function?
the same as the point that has a local maximum or minimum?

Answer 27

the local minimum

Answer 28

and also the local maximum, because at both the direction of motion is changed

Answer 29

Let me know when you are back!

Answer 30

Sorry I am trying to figure this out

Answer 31

Im still a bit confused

Answer 32

do we just derive it twice and sub in 3? or some other value?

Answer 33

I am sure that we should derive it twice but I am not sure which point would we substitute with in them!!!

@jagr2713
If you could help here, I would be grateful!

Answer 34

would it be the point on the graph that is the minimum?

Answer 35

I am not sure: maximum or minimum or both!

Answer 36

I don't do calculus yet bud D:

Answer 37

If you could help here @triciaal, I would be grateful!

Answer 38

I think i figured this one out!

Answer 39

how?

Answer 40

The particle will reverse directions when the velocity is equal to zero, the first derivative of this function is the velocity equation (as stated above). so we need to find what the values of x are when y=0. Once solved, we find these to be x=3 and x=5 (as i had found before). we then use these to find the position by plugging them into the original equation. s(t) = 3^3 – 12 * 3^2 + 45 * 3 + 4 = 58
s(t) = 5^3 – 12 * 5^2 + 45 * 5 + 4 = 54
we then get 58 and 54 as the positions and we derive the velocity equation (as i found above) and get a = 6 * t – 24
we subsititue 3 and 5 in and get -6 and 6

Answer 41

(sorry that took me a little while to type)

Answer 42

No problem.

Answer 43

"The particle will reverse directions when the velocity is equal to zero,"
The velocity may be zero (as a stop) and return to be greater than zero again in the same direction!

Answer 44

but the graph is a parabola

Answer 45

So what you mean?

Answer 46

Let me know when you are back!