What is the average speed of an object that is moving at $18 m/s$ at $t=0$ and accelerates at a rate of $a(t) =4t-8$ on $t in [0,2]$ ?

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Answer 1 · 2016-06-28T14:49:32

$v_a=3.5" "m/s$

$"Strategy..."$
$diamond " find v(t) function"$
$diamond" use average value theorem"$

$diamond" --> "v(t)=int a(t) d t$

$v(t)=int (4t-8)d t$

$v(t)=4*1/2*t^2-8*t$

$v(t)=2t^2-8t+C$

$"for "t=0" ; " v=18" "m/s" ; ""then "C=18$

$v(t)=2t^2-8t+18$

$diamond" -->"v_a=1/(2-0)int_0^2v(t) d t$

$v_a=1/2int_0^2 (2t^2-8t+18)d t$

$v_a=1/2[|2*1/3*t^3-8*1/2*t^2+18*t|_0^2]$

$v_a=1/2[|2/3t^3-4t^2+18t|_0^2]$

$v_a=1/2[(2/3*2^3-8*2^2+18*2)-(0)]$

$v_a=1/2[16/3-32+36]$

$v=a=1/2[16/3+4]=1/2[(16+12)/4]=1/2[28/4]=1/2*7$

$v_a=3.5" "m/s$

What is the average speed of an object that is moving at 18 m/s18 m/s at t=0t=0 and accelerates at a rate of a(t) =4t-8a(t) =4t-8 on t in [0,2]t in [0,2]?