A particle is moving with the given data. Find the position of the particle. a(t) = t^2 − 9t + 7, s(0) = 0, s(1) = 20?

We need to find s(t)

Now, v(t) = s'(t) and a(t) = v'(t)

so, given a(t) we need to integrate twice to get our position function.

Now, ∫ t² - 9t + 7 dt => t³/3 - 9t²/2 + 7t + C

Hence, v(t) = t³/3 - 9t²/2 + 7t + C

Now, ∫ t³/3 - 9t²/2 + 7t + C dt => t⁴/12 - 3t³/2 + 7t²/2 + Ct + D

i.e. s(t) = t⁴/12 - 3t³/2 + 7t²/2 + Ct + D

If s(0) = 0 then,

0 = D

If s(1) = 20 then,

20 = 1/12 - 3/2 + 7/2 + C

i.e. C = 215/12

Then, s(t) = t⁴/12 - 3t³/2 + 7t²/2 + 215t/12

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a(t) = t^2 - 9t + 7

Integrating gives:

v(t) = (1/3)t^3 - (9/2)t^2 + 7t + C

Integrating again gives:

s(t) = (1/12)t^4 - (3/2)t^3 + (7/2)t^2 + (c1)t + c2

s(0) = 0 implies:

0 = c2

s(1) = 20 implies:

20 = (1/12) - (3/2) + (7/2) + c1

c1 = 215/12

s(t) = (1/12)t^4 - (3/2)t^3 + (7/2)t^2 + 215/12