An object has a mass of #8 kg#. The object's kinetic energy uniformly changes from #24 KJ# to # 64KJ# over #t in [0, 3 s]#. What is the average speed of the object?

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Answer 1 · 2017-06-21T12:56:02

The average speed is #=103.9ms^-1#

The kinetic energy is

#KE=1/2mv^2#

The mass is #=8kg#

The initial velocity is #=u_1ms^-1#

The final velocity is #=u_2 ms^-1#

The initial kinetic energy is #1/2m u_1^2=24000J#

The final kinetic energy is #1/2m u_2^2=64000J#

Therefore,

#u_1^2=2/8*24000=6000m^2s^-2#

and,

#u_2^2=2/8*64000=16000m^2s^-2#

The graph of #v^2=f(t)# is a straight line

The points are #(0,6000)# and #(3,16000)#

The equation of the line is

#v^2-6000=(16000-6000)/3t#

#v^2=3333.3t+6000#

So,

#v=sqrt((3333.3t+6000)#

We need to calculate the average value of #v# over #t in [0,3]#

#(3-0)bar v=int_0^3sqrt((3333.3t+6000))dt#

#3 barv=[((3333.3t+6000)^(3/2)/(3/2*3333.3)]_0^3#

#=((3333.3*3+6000)^(3/2)/(5000))-((3333.3*0+6000)^(3/2)/(5000))#

#=16000^(3/2)/5000-6000^(3/2)/5000#

#=311.8#

So,

#barv=311.8/3=103.9ms^-1#

The average speed is #=103.9ms^-1#