What is the average speed of an object that is moving at $3 m/s$ at $t=0$ and accelerates at a rate of $a(t) =t-1$ on $t in [0,2]$ ?

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Answer 1 · 2016-05-22T09:04:54

$v_a=8/3" "m/s$

$v(t)=int a(t) d t$

$v(t)=int (t-1)d t$

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

$"for t=0 ; v=3 ; C=3"$

$v(t)=1/2 t^2-t+3$

$"1-Solution using ' the average value' theorem :"$

$v_a=1/(2-0) int _0^2(1/2t^2-t+3)$

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

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

$v_a=1/2[4/3-2+6]$

$v_a=1/2[4/3+4]$

$v_a=1/2[16/3]$

$v_a=16/6$

$v_a=8/3" "m/s$

$"2-Solution using definition of average velocity"$

$v_a=(Delta x)/(Delta t)=("displacement")/("time")$

$Delta x=int _0^2 v(t) d t=int _0^2 (1/2t^2-t+3)d x$

$Delta x=[|1/2*1/3*t^3-1/2 t^2-3t|_0^2]=[(1/2*1/3*2^3-1/2*2^2+3*2)-0]$

$Delta x=4/3-2+6" ; "Delta x=4/3+4" ; "Delta x=16/3" "m$

$v_a=(16/3)/2$

$v_a=16/6$

$v_a=8/3" "m/s$

What is the average speed of an object that is moving at 3 m/s3 m/s at t=0t=0 and accelerates at a rate of a(t) =t-1a(t) =t-1 on t in [0,2]t in [0,2]?