An object has a mass of 4 kg. The object's kinetic energy uniformly changes from 42 KJ to 320 KJ over t in [0, 12 s]. What is the average speed of the object?

1 Answer
Narad T.
May 6, 2017

The average speed is =292.4ms^-1

Explanation:

The kinetic energy is

KE=1/2mv^2

mass is =4kg

The initial velocity is =u_1

1/2m u_1^2=42000J

The final velocity is =u_2

1/2m u_2^2=320000J

Therefore,

u_1^2=2/4*42000=21000m^2s^-2

and,

u_2^2=2/4*320000=160000m^2s^-2

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

The points are (0,21000) and (12,160000)

The equation of the line is

v^2-21000=(160000-21000)/12t

v^2=11583.3t+21000

So,

v=sqrt((11583.3t+21000)

We need to calculate the average value of v over t in [0,12]

(12-0)bar v=int_0^12sqrt((11583.3t+21000))dt

12 barv=[((11583.3t+21000)^(3/2)/(3/2*11583.3)]_0^12

=((11583.3*12+21000)^(3/2)/(17375))-((1583.3*0+21000)^(3/2)/(17375))

=160000^(3/2)/17375-21000^(3/2)/17375

=3508.3

So,

barv=3508.3/12=292.4ms^-1

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