An object has a mass of #6 kg#. The object's kinetic energy uniformly changes from #66 KJ# to # 225 KJ# over #t in [0, 8 s]#. What is the average speed of the object?

Question

An object has a mass of #6 kg#. The object's kinetic energy uniformly changes from #66 KJ# to # 225  KJ# over #t in [0,  8  s]#.  What is the average speed of the object?

William Audy · Accepted Answer

The average speed is #=217.3ms^-1#

The kinetic energy is

#KE=1/2mv^2#

The mass is #=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=66000J#

The final kinetic energy is #1/2m u_2^2=225000J#

Therefore,

#u_1^2=2/6*66000=22000m^2s^-2#

and,

#u_2^2=2/6*225000=75000m^2s^-2#

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

The points are #(0,22000)# and #(8,75000)#

The equation of the line is

#v^2-22000=(75000-22000)/8t#

#v^2=6625t+22000#

So,

#v=sqrt((6625t+22000)#

We need to calculate the average value of #v# over #t in [0,8]#

#(8-0)bar v=int_0^8sqrt(6625t+22000))dt#

#8 barv=[((6625t+22000)^(3/2)/(3/2*6625)]_0^8#

#=((6625*8+22000)^(3/2)/(9937.5))-((6625*0+22000)^(3/2)/(9937.5))#

#=75000^(3/2)/9937.5-22000^(3/2)/9937.5#

#=1738.5#

So,

#barv=1738.5/8=217.3ms^-1#

The average speed is #=217.3ms^-1#

Isabella Alvarez · Answer

undefined To find the average speed of the object, we need to calculate the change in kinetic energy and then use it to determine the average speed. First, we find the change in kinetic energy:

Change in kinetic energy = Final kinetic energy - Initial kinetic energy
                           = 225 kJ - 66 kJ
                           = 159 kJ

Then, we can use the formula for kinetic energy:

Kinetic energy = (1/2) * mass * velocity^2

Solving for velocity:

Velocity = sqrt((2 * Kinetic energy) / mass)

Substituting the given values:

Velocity = sqrt((2 * 159 kJ) / 6 kg)
         ≈ sqrt(53 kJ / kg)

Now, we need to convert kJ/kg to m^2/s^2 (since 1 kJ/kg equals 1 m^2/s^2):

1 kJ/kg = 1 m^2/s^2

So, the velocity is approximately the square root of 53 m^2/s^2. This gives us the average speed of the object.