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]#?

1 Answer
Jun 28, 2016

Answer:

#v_a=3.5" "m/s#

Explanation:

#"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#