Question

A particle moves along a horizontal line. Its position function is s(t) for t is greater than or equal to 0. For each problem, find the velocity function v(t) and the acceleration function a(t).
1. s(t) = -t^4 + 12t^3
2. s(t) = -t^4 + 8t^3

Answer 1

Answer:

1) Velocity

$v(t) = -4\cdot t^{3}+36\cdot t^{2}$

Acceleration

$a(t) = -12\cdot t^{2}+72\cdot t$

2) Velocity

$v(t) = -4\cdot t^{3}+24\cdot t^{2}$

Acceleration

$a(t) = -12\cdot t^{2}+48\cdot t$

Step-by-step explanation:

From Physics we remember that velocity ($v(t)$) and acceleration ($a(t)$) are the first and second derivatives of the function position in time. That is:

1) Let $s(t) = -t^{4}+12\cdot t^{3}$, where $t \ge 0$. The functions velocity and aceleration are, respectively:

Velocity

$v(t) = -4\cdot t^{3}+36\cdot t^{2}$

Acceleration

$a(t) = -12\cdot t^{2}+72\cdot t$

2) Let $s(t) = -t^{4}+8\cdot t^{3}$, where $t\ge 0$. The functions velocity and acceleration are, respectively:

Velocity

$v(t) = -4\cdot t^{3}+24\cdot t^{2}$

Acceleration

$a(t) = -12\cdot t^{2}+48\cdot t$