An object has a mass of #4 kg#. The object's kinetic energy uniformly changes from #18 KJ# to # 42KJ# over #t in [0, 9 s]#. What is the average speed of the object?

Question

An object has a mass of #4 kg#. The object's kinetic energy uniformly changes from #18 KJ# to # 42KJ# over #t in [0,  9  s]#.  What is the average speed of the object?

Charles Aeschliman · Accepted Answer

The average speed is #=121.6ms^-1#

The kinetic energy is

#KE=1/2mv^2#

The mass is #=4kg#

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=42000J#

Therefore,

#u_1^2=2/4*18000=9000m^2s^-2#

and,

#u_2^2=2/4*42000=21000m^2s^-2#

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

The points are #(0,9000)# and #(9,21000)#

The equation of the line is

#v^2-9000=(21000-9000)/9t#

#v^2=1333.3t+9000#

So,

#v=sqrt((1333.3t+9000)#

We need to calculate the average value of #v# over #t in [0,9]#

#(9-0)bar v=int_0^9(sqrt(1333.3t+9000))dt#

#9 barv=[((1333.3t+9000)^(3/2)/(3/2*1333.3))]_0^9#

#=((1333.3*9+9000)^(3/2)/(2000))-((1333.3*0+9000)^(3/2)/(2000))#

#=21000^(3/2)/2000-9000^(3/2)/2000#

#=1094.7#

So,

#barv=1094.7/9=121.6ms^-1#

The average speed is #=121.6ms^-1#

Charlotte Aaron · Answer

undefined To find the average speed of the object, we need to calculate the change in kinetic energy and divide it by the mass of the object. The change in kinetic energy is the difference between the final kinetic energy and the initial kinetic energy. Then, we divide this by the time interval over which the change occurs.

Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Change in kinetic energy = 42 kJ - 18 kJ = 24 kJ

Average speed = Change in kinetic energy / Mass / Time
Average speed = (24 kJ) / (4 kg) / (9 s) = 0.667 m/s