An object has a mass of 5 kg. The object's kinetic energy uniformly changes from 15 KJ to 135 KJ over t in [0, 9 s]. What is the average speed of the object?

1 Answer
Narad T.
Jun 10, 2017

The average speed is =167.8ms^-1

Explanation:

The kinetic energy is

KE=1/2mv^2

The mass is =5kg

The initial velocity is =u_1

The initial kinetic energy is 1/2m u_1^2=15000J

The final velocity is =u_2

The final kinetic energy is 1/2m u_2^2=135000J

Therefore,

u_1^2=2/5*15000=6000m^2s^-2

and,

u_2^2=2/5*135000=54000m^2s^-2

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

The points are (0,6000) and (9,54000)

The equation of the line is

v^2-6000=(54000-6000)/9t

v^2=5333.3t+6000

So,

v=sqrt((5333.3t+6000)

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

(9-0)bar v=int_0^9sqrt((5333.3t+6000))dt

9 barv=[((5333.3t+6000)^(3/2)/(3/2*5333.3)]_0^9

=((5333.3*9+6000)^(3/2)/(8000))-((5333.3*0+6000)^(3/2)/(8000))

=54000^(3/2)/8000-6000^(3/2)/8000

=1510.5

So,

barv=1510.5/9=167.8ms^-1

The average speed is =167.8ms^-1

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