An object has a mass of #6 kg#. The object's kinetic energy uniformly changes from #480 KJ# to #12 KJ# over #t in [0, 8 s]#. What is the average speed of the object?

Answer 1

The average speed is #=272.4ms^-1#

The kinetic energy is

#KE=1/2mv^2#
The mass is #m=6kg#
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=480000J#
The final kinetic energy is #1/2m u_2^2=12000J#

Therefore,

#u_1^2=2/6*480000=160000m^2s^-2#

and,

#u_2^2=2/6*12000=4000m^2s^-2#
The graph of #v^2=f(t)# is a straight line
The points are #(0,160000)# and #(8,4000)#

The equation of the line is

#v^2-160000=(4000-160000)/8t#
#v^2=-19500t+160000#

So,

#v=sqrt(-19500t+160000)#
We need to calculate the average value of #v# over #t in [0,8]#
#(8-0)bar v=int_0^8(sqrt(-19500t+160000))dt#
#8 barv= [(-19500t+160000)^(3/2)/(-3/2*19500)] _( 0) ^ (8)#
#=((-19500*8+160000)^(3/2)/(-29250))-((-19500*0+160000)^(3/2)/(-29250))#
#=160000^(3/2)/29250-4000^(3/2)/29250#
#=2179.4#

So,

#barv=2179.4/8=272.4ms^-1#
The average speed is #=272.4ms^-1#
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Answer 2

To find the average speed of the object, divide the total distance traveled by the total time taken. Given that the kinetic energy changes uniformly, the average kinetic energy over the interval is simply the average of the initial and final kinetic energies. Using the kinetic energy formula, we can find the initial and final velocities of the object. Then, using the formula for average speed, we can calculate it.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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