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#