# An object has a mass of #4 kg#. The object's kinetic energy uniformly changes from #12 KJ# to # 36 KJ# over #t in [0, 12 s]#. What is the average speed of the object?

##### 2 Answers

The average speed is

#### Explanation:

The kinetic energy is

The mass is

The initial velocity is

The final velocity is

The initial kinetic energy is

The final kinetic energy is

Therefore,

and,

The graph of

The points are

The equation of the line is

So,

We need to calculate the average value of

So,

The average speed is

Kinetic energy

Given

Let initial velocity be

The final velocity be

Initial kinetic energy

#1/2m u^2=12000J#

Final kinetic energy is

#1/2m v^2=36000J#

As kinetic energy changes uniformly, the graph of

The points on the graph are

The equation of the line is of the type

#1/2m(v(t))^2=(36000-12000)/12t+12000#

or#1/2xx4(v(t))^2=2000t+12000#

#=>(v(t))^2=1000t+6000#

#=>v(t)=sqrt(1000t+6000)#

We know that velocity can be written as

#=>(ds(t))/dt=sqrt(1000t+6000)#

We need to calculate total distance traveled in time

#:.s=int_0^12sqrt(1000t+6000)cdotdt#

#=>s=[(1000t+6000)^(3/2)/(3/2xx1000)] _( 0) ^ (12) #

#=>s=1/1500[(1000xx12+6000)^(3/2)-(1000xx0+6000)^(3/2)]#

#=>s=1/1500(18000^(3/2)-6000^(3/2))#

#=>s=1300.13m#

We know that