An object has a mass of #6 kg#. The object's kinetic energy uniformly changes from #18 KJ# to # 64KJ# over #t in [0,12s]#. 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 #18 KJ# to # 64KJ# over #t in [0,12s]#.  What is the average speed of the object?

Elena Albarran · Accepted Answer

The average speed is #=115.3ms^-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=18000J#

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

Therefore,

#u_1^2=2/6*18000=6000m^2s^-2#

and,

#u_2^2=2/6*64000=21333.3m^2s^-2#

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

The points are #(0,6000)# and #(12,21333.3)#

The equation of the line is

#v^2-6000=(21333.3-6000)/12t#

#v^2=1277.8t+6000#

So,

#v=sqrt(1277.8t+6000)#

We need to calculate the average value of #v# over #t in [0,12]#

#(12-0)bar v=int_0^12(sqrt(1277.8t+6000))dt#

#12 barv= [(1277.8t+6000)^(3/2)/(3/2*1277.8)] _( 0) ^ (12)#

#=((1277.8*12+6000)^(3/2)/(1916.7))-((1277.8*0+6000)^(3/2)/(1916.7))#

#=21333.3^(3/2)/1916.7-6000^(3/2)/1916.7#

#=1383.2#

So,

#barv=1383.2/12=115.3ms^-1#

The average speed is #=115.3ms^-1#

Nicholas Agrusa · Answer

undefined To find the average speed of the object, you need to calculate the change in kinetic energy and then use it along with the given time interval to find the average speed. The change in kinetic energy is 64 KJ - 18 KJ = 46 KJ. To convert this change in kinetic energy to joules (J), multiply by 1000, because 1 KJ = 1000 J. So, the change in kinetic energy is 46,000 J. Now, you can use the formula for kinetic energy: KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the speed. Rearranging the formula to solve for speed gives: v = sqrt(2KE / m). Plugging in the values, you get: v = sqrt(2 * 46000 J / 6 kg). After calculating this expression, you will get the average speed of the object.