An object has a mass of #4 kg#. The object's kinetic energy uniformly changes from #64 KJ# to # 160 KJ# over #t in [0, 12 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 #64 KJ# to # 160  KJ# over #t in [0,  12  s]#.  What is the average speed of the object?

Adrian Ayala · Accepted Answer

The average speed is #=234.8ms^-1#

The kinetic energy is

#KE=1/2mv^2#

mass is #=4kg#

The initial velocity is #=u_1#

#1/2m u_1^2=64000J#

The final velocity is #=u_2#

#1/2m u_2^2=160000J#

Therefore,

#u_1^2=2/4*64000=32000m^2s^-2#

and,

#u_2^2=2/4*160000=80000m^2s^-2#

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

The points are #(0,32000)# and #(12,80000)#

The equation of the line is

#v^2-32000=(80000-32000)/12t#

#v^2=4000t+32000#

So,

#v=sqrt((4000t+32000)#

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

#(12-0)bar v=int_0^12sqrt((4000t+32000))dt#

#12 barv=[((4000t+32000)^(3/2)/(3/2*4000)]_0^12#

#=((4000*12+32000)^(3/2)/(6000))-((4000*0+32000)^(3/2)/(6000))#

#=80000^(3/2)/6000-32000^(3/2)/6000#

#=2817.2#

So,

#barv=2817.2/12=234.8ms^-1#

Amelia Adams · Answer

undefined To find the average speed of the object, we need to use the formula:

Average speed = Total distance traveled / Total time taken

In this case, since the object's mass is given, we'll use the change in kinetic energy to calculate the total distance traveled. We'll also use the given time interval.

First, we find the change in kinetic energy:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Change in kinetic energy = 160 kJ - 64 kJ = 96 kJ

Next, we convert the change in kinetic energy from kilojoules to joules:
Change in kinetic energy = 96 kJ * 1000 = 96000 J

Since kinetic energy is given by the formula KE = (1/2) * m * v^2, where m is mass and v is velocity, we can rearrange the formula to solve for velocity:
v = sqrt((2 * KE) / m)

Now, we plug in the values:
v = sqrt((2 * 96000 J) / 4 kg)

v = sqrt(48000) m/s

v ≈ 219.088 m/s

Therefore, the average speed of the object is approximately 219.088 meters per second.