An object has a mass of 4 kg. The object's kinetic energy uniformly changes from 18 KJ to 48KJ over t in [0, 9 s]. What is the average speed of the object?

1 Answer
Narad T.
Oct 28, 2017

The average speed is =127.3ms^-1

Explanation:

The kinetic energy is

KE=1/2mv^2

The mass is m=4kg

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=18000J

The final kinetic energy is 1/2m u_2^2=48000J

Therefore,

u_1^2=2/4*18000=9000m^2s^-2

and,

u_2^2=2/4*48000=24000m^2s^-2

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

The points are (0,9000) and (9,24000)

The equation of the line is

v^2-9000=(24000-9000)/9t

v^2=1666.7t+9000

So,

v=sqrt((1666.7t+9000)

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

(9-0)bar v=int_0^12(sqrt(1666.7t+9000))dt

9 barv=[((1666.7t+9000)^(3/2)/(3/2*1666.7))]_0^9

=((1666.7*9+9000)^(3/2)/(2500))-((-1666.7*0+9000)^(3/2)/(2500))

=24000^(3/2)/2500-9000^(3/2)/2500

=1145.7

So,

barv=1145.7/9=127.3ms^-1

The average speed is =127.3ms^-1

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