An object has a mass of #2 kg#. The object's kinetic energy uniformly changes from #64 KJ# to #28 KJ# over #t in [0, 15 s]#. What is the average speed of the object?
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"What does Hess's law say about the enthalpy of a reaction?"
The average speed is #=319.6ms^-1#
The kinetic energy is
#KE=1/2mv^2#
The mass is #m=2kg#
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=64000J#
The final kinetic energy is #1/2m u_2^2=28000J#
Therefore,
#u_1^2=2/2*64000=64000m^2s^-2#
and,
#u_2^2=2/2*28000=28000m^2s^-2#
The graph of #v^2=f(t)# is a straight line
The points are #(0,64000)# and #(15,28000)#
The equation of the line is
#v^2-64000=(28000-64000)/15t#
#v^2=-2400t+64000#
So,
#v=sqrt(-2400t+64000)#
We need to calculate the average value of #v# over #t in [0,15]#
#(15-0)bar v=int_0^15(sqrt(-2400t+64000))dt#
#15 barv= (-2400t+64000)^(3/2)/(3/2*-2400)| _( 0) ^ (15) #
#=((-2400*15+64000)^(3/2)/(-2400))-((-2400*0+64000)^(3/2)/(-2400))#
#=64000^(3/2)/2400-28000^(3/2)/2400#
#=4794#
So,
#barv=4794/15=319.6ms^-1#
The average speed is #=319.6ms^-1#
