An object has a mass of #2 kg#. The object's kinetic energy uniformly changes from #64 KJ# to #72 KJ# over #t in [0, 3 s]#. What is the average speed of the object?

Question

An object has a mass of #2 kg#. The object's kinetic energy uniformly changes from #64 KJ# to #72  KJ# over #t in [0,  3  s]#.  What is the average speed of the object?

Olivia Askew · Accepted Answer

The average speed is #=260.7ms^-1#

The kinetic energy is

#KE=1/2mv^2#

The mass is #=2kg#

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=64000J#

The final kinetic energy is #1/2m u_2^2=72000J#

Therefore,

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

and,

#u_2^2=2/2*72000=72000m^2s^-2#

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

The points are #(0,64000)# and #(3,72000)#

The equation of the line is

#v^2-64000=(72000-64000)/3t#

#v^2=2666.7t+64000#

So,

#v=sqrt((2666.7t+64000)#

We need to calculate the average value of #v# over #t in [0,3]#

#(3-0)bar v=int_0^3(sqrt(2666.7t+64000))dt#

#3 barv=[((2666.7t+64000)^(3/2)/(3/2*2666.7))]_0^3#

#=((2666.7*3+64000)^(3/2)/(4000))-((2666.7*0+64000)^(3/2)/(4000))#

#=72000^(3/2)/4000-64000^(3/2)/4000#

#=782.2#

So,

#barv=782.2/3=260.7ms^-1#

The average speed is #=260.7ms^-1#

Xavier Adame · Answer

undefined The average speed of the object can be calculated using the formula:

Average speed = Total distance traveled / Total time taken

Since the object's kinetic energy changes uniformly, we can use the change in kinetic energy to find the total distance traveled.

First, we need to find the change in kinetic energy:

Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Change in kinetic energy = 72 kJ - 64 kJ
Change in kinetic energy = 8 kJ

Now, we'll convert the change in kinetic energy from kJ to J:
Change in kinetic energy = 8 kJ * 1000 J/kJ
Change in kinetic energy = 8000 J

Next, we'll use the formula for kinetic energy:
Kinetic energy = (1/2) * mass * velocity^2

We'll rearrange the formula to solve for velocity:
Velocity = sqrt((2 * kinetic energy) / mass)

We'll plug in the values:
Velocity = sqrt((2 * 8000 J) / 2 kg)
Velocity = sqrt(8000 J / 2 kg)
Velocity = sqrt(4000 m^2/s^2)
Velocity = 63.25 m/s

Now, we'll use the average speed formula:
Average speed = Total distance traveled / Total time taken
Average speed = velocity * t
Average speed = 63.25 m/s * 3 s
Average speed = 189.75 m/s

So, the average speed of the object is 189.75 m/s.