An object has a mass of #8 kg#. The object's kinetic energy uniformly changes from #240 KJ# to # 64KJ# over #t in [0, 3 s]#. What is the average speed of the object?

Question

An object has a mass of #8 kg#. The object's kinetic energy uniformly changes from #240 KJ# to # 64KJ# over #t in [0,  3  s]#.  What is the average speed of the object?

Nathan Ashcroft · Accepted Answer

The average speed is #=192ms^-1#

The kinetic energy is

#KE=1/2mv^2#

The mass is #=8kg#

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=240000J#

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

Therefore,

#u_1^2=2/8*240000=60000m^2s^-2#

and,

#u_2^2=2/8*64000=16000m^2s^-2#

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

The points are #(0,60000)# and #(3,16000)#

The equation of the line is

#v^2-60000=(16000-60000)/3t#

#v^2=-14666.7t+60000#

So,

#v=sqrt((-14666.7t+60000)#

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

#(3-0)bar v=int_0^3sqrt(-14666.7t+60000))dt#

#3 barv=[((-14666.7t+60000)^(3/2)/(-3/2*14666.7)]_0^3#

#=((-14666.7*3+60000)^(3/2)/(22000))-((14666.7*0+60000)^(3/2)/(22000))#

#=60000^(3/2)/22000-16000^(3/2)/22000#

#=576#

So,

#barv=576/3=192ms^-1#

The average speed is #=192ms^-1#

Isabella Alvarez · Answer

undefined To find the average speed of the object, we need to calculate the total distance traveled by the object over the given time interval and then divide it by the total time.

First, we find the initial and final velocities using the kinetic energy formula:

Initial kinetic energy (KE_initial) = 240 kJ
Final kinetic energy (KE_final) = 64 kJ

We know that kinetic energy (KE) is given by the formula:
KE = (1/2) * mass * velocity^2

Using the formula, we can solve for the initial and final velocities:

KE_initial = (1/2) * mass * initial_velocity^2
240 kJ = (1/2) * 8 kg * initial_velocity^2
initial_velocity = √(2 * 240 kJ / 8 kg)

Similarly,
KE_final = (1/2) * mass * final_velocity^2
64 kJ = (1/2) * 8 kg * final_velocity^2
final_velocity = √(2 * 64 kJ / 8 kg)

Next, we find the average speed (v_avg) using the formula:
v_avg = total_distance / total_time

Since the object's velocity may not be constant, we can use the average of the initial and final velocities as the object's average velocity.

v_avg = (initial_velocity + final_velocity) / 2

Finally, we can plug in the values to find the average speed of the object.