An object has a mass of #9 kg#. The object's kinetic energy uniformly changes from #135 KJ# to # 45 KJ# over #t in [0, 4 s]#. What is the average speed of the object?

Question

An object has a mass of #9 kg#. The object's kinetic energy uniformly changes from #135 KJ# to # 45  KJ# over #t in [0,  4  s]#.  What is the average speed of the object?

Benjamin Asiedu · Accepted Answer

Average Speed #=136.6025404" "#m/sec

The given data: Mass #m=9" "#kg Kinetic Energy initial #KE_i=135000" "#joule Kinetic Energy final #KE_f=45000" "#joule initial time #t_0=0" "#sec final time #t_1=4" "#sec Compute for initial velocity #v_i# #KE_i=1/2mv_i^2# #135000=1/2(9)v_i^2# #v_i^2=(2(135000))/9# #v_i=sqrt(30000)# #v_i=173.2050808" "#m/sec Compute for final velocity #v_f# #KE_f=1/2mv_f^2# #45000=1/2(9)v_f^2# #v_f^2=(2(45000))/9# #v_f=sqrt(10000)# #v_f=100" "#m/sec Solve for the total distance traveled by the object using the following: #v_f^2=v_i^2+2a*s" "#first equation and #v_f=v_i+a*t" "#second equation #v_f^2-v_i^2=2a*s" "#from the first equation #color(red)((v_f+v_i)(v_f-v_i)=2*a*s)# #color(blue)((v_f-v_i)=a*t" ")#from the second equation Divide the first equation by the second equation #color(red)((v_f+v_i)cancel(v_f-v_i))/color(blue)(cancel(v_f-v_i))=color(red)(2*cancela*s)/color(blue)(cancela*t)# #v_f+v_i=(2s)/t# #s=1/2*t(v_f+v_i)# #s=1/2*(4)(100+173.2050808)# #s=546.4101616" "#m Average speed #=("distance traveled")/("elapsed time")# Average speed #=546.4101616/4# Average speed #=136.6025404" "#m/sec God bless....I hope the explanation is useful.

Xavier Adame · Answer

undefined To find the average speed of the object, you need to determine the total distance traveled by the object during the time interval [0, 4 s].

First, calculate the change in kinetic energy:

ΔKE = 135 KJ - 45 KJ = 90 KJ

Next, use the work-energy principle, which states that the work done on an object equals the change in kinetic energy:

Work = ΔKE

Work = Force × Distance

Since the force acting on the object is constant, you can use the equation:

Work = Force × Distance = ΔKE

Rearrange the equation to solve for distance:

Distance = ΔKE / Force

Now, you need to determine the force acting on the object. Since the object's mass is given as 9 kg, you can use Newton's second law, which states that force equals mass times acceleration:

Force = mass × acceleration

Acceleration can be calculated using the change in kinetic energy and the time interval:

ΔKE = (1/2) × mass × (final velocity^2 - initial velocity^2)

ΔKE = 90 KJ = (1/2) × 9 kg × (v_final^2 - v_initial^2)

You're given that the time interval is 4 seconds. Therefore, you can calculate the initial and final velocities:

Initial velocity, v_initial = Distance / time = 0 / 4 = 0 m/s

Final velocity, v_final = Distance / time = Distance / 4

Now, substitute these values back into the equation for ΔKE:

90 KJ = (1/2) × 9 kg × [(Distance / 4)^2 - (0 m/s)^2]

Solve for Distance:

90 KJ = (1/2) × 9 kg × (Distance^2 / 16)

Distance^2 = (90 KJ * 2 * 16) / 9 kg

Distance = sqrt((90 KJ * 2 * 16) / 9 kg)

Once you have the distance, divide it by the total time to find the average speed of the object.