What is the average speed of an object that is moving at $3 \frac{m}{s}$ $3 m/s$ at $t = 0$ $t=0$ and accelerates at a rate of $a (t) = 9 - \frac{t^{2}}{3}$ $a(t) =9-t^2/3$ on $t \in [3, 5]$ $t in [3,5]$ ?

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What is the average speed of an object that is moving at $3 \frac{m}{s}$ $3 m/s$ at $t = 0$ $t=0$ and accelerates at a rate of $a (t) = 9 - \frac{t^{2}}{3}$ $a(t) =9-t^2/3$ on $t \in [3, 5]$ $t in [3,5]$ ?

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Answer 1 · 2017-03-30T03:29:15

$31 \frac{4}{9}$ $31 4/9$ m/s

Explanation:

$a (t) = 9 - \frac{t^{2}}{3}$ $a(t) = 9 -t^2/3$

Speed, $v = \int a (t) = \int 9 - \frac{t^{2}}{3} = 9 t - \frac{t^{3}}{9} + c$ $v = int a(t) = int 9 -t^2/3 =9t - t^3/9 + c$

$t = 0, v = 3$ $t = 0, v = 3$ m/s $\to c = 3$ $-> c = 3$ , therefore,
$v = 9 t - \frac{t^{3}}{9} + 3$ $v = 9t - t^3/9 + 3$

Displacement,
$s = \int v (t)$ $s = int v(t)$ ,

$s_{3, 5} = \int_{3}^{5} v (t) = \int_{3}^{5} 9 t - \frac{t^{3}}{9} + 3 = {[\frac{9}{2} t^{2} - \frac{t^{4}}{36} + 3 t]}_{3}^{2}$ $s_(3,5) = int_3^5 v(t) = int_3^5 9t - t^3/9 + 3 = [9/2 t^2 - t^4/36 + 3t]_3^5$

$s_{3, 5} = [\frac{9}{2} {(5)}^{2} - \frac{{(5)}^{4}}{36} + 3 (5)] - [\frac{9}{2} {(3)}^{2} - \frac{{(3)}^{4}}{36} + 3 (3)]$ $s_(3,5) = [9/2 (5)^2 - (5)^4/36 + 3(5)] - [9/2 (3)^2 - (3)^4/36 + 3(3)]$

$s_{3, 5} = [\frac{225}{2} - \frac{625}{36} + 15] - [\frac{81}{2} - \frac{81}{36} + 9]$ $s_(3,5) = [225/2 - 625/36 + 15] - [81/2 - 81/36 + 9]$

$s_{3, 5} = \frac{225}{2} - \frac{81}{2} - \frac{625}{36} + \frac{81}{36} + 15 - 9$ $s_(3,5) = 225/2 - 81/2- 625/36 + 81/36 + 15- 9$

$s_{3, 5} = \frac{144}{2} - \frac{544}{36} + 6$ $s_(3,5) = 144/2- 544/36 + 6$

$s_{3, 5} = 72 - \frac{136}{9} + 6 = \frac{566}{9}$ $s_(3,5) = 72 - 136/9 + 6 = 566/9$

Average speed = displacement/time $= \frac{566}{9} \div 2 = \frac{566}{18} = 31 \frac{4}{9}$ $=566/9-:2 = 566/18 = 31 4/9$ m/s

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What is the average speed of an object that is moving at $3 \frac{m}{s}$ $3 m/s$ at $t = 0$ $t=0$ and accelerates at a rate of $a (t) = 9 - \frac{t^{2}}{3}$ $a(t) =9-t^2/3$ on $t \in [3, 5]$ $t in [3,5]$ ?

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What is the average speed of an object that is moving at $3 \frac{m}{s}$ $3 m/s$ at $t = 0$ $t=0$ and accelerates at a rate of $a (t) = 9 - \frac{t^{2}}{3}$ $a(t) =9-t^2/3$ on $t \in [3, 5]$ $t in [3,5]$ ?