An object has a mass of 12 kg. The object's kinetic energy uniformly changes from 96 KJ to 160 KJ over t in [0, 4 s]. What is the average speed of the object?

1 Answer
Narad T.
Jun 19, 2017

The average speed is =145.7ms^-1

Explanation:

The kinetic energy is

KE=1/2mv^2

The mass is =12kg

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=96000J

The final kinetic energy is 1/2m u_2^2=160000J

Therefore,

u_1^2=2/12*96000=16000m^2s^-2

and,

u_2^2=2/12*160000=26666.7m^2s^-2

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

The points are (0,16000) and (4,26666.7)

The equation of the line is

v^2-16000=(26666.7-16000)/4t

v^2=2666.7t+16000

So,

v=sqrt((2666.7t+16000)

We need to calculate the average value of v over t in [0,4]

(4-0)bar v=int_0^4sqrt((2666.7t+16000))dt

4 barv=[((2666.7t+16000)^(3/2)/(3/2*2666.7)]_0^4

=((2666.7*4+16000)^(3/2)/(4000))-((2666.7*0+16000)^(3/2)/(4000))

=26666.7^(3/2)/4000-16000^(3/2)/4000

=582.7

So,

barv=582.7/4=145.7ms^-1

The average speed is =145.7ms^-1

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