An object has a mass of 4 kg. The object's kinetic energy uniformly changes from 18 KJ to 42KJ over t in [0, 9 s]. What is the average speed of the object?

1 Answer
Narad T.
Aug 13, 2017

The average speed is =121.6ms^-1

Explanation:

The kinetic energy is

KE=1/2mv^2

The mass is =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=42000J

Therefore,

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

and,

u_2^2=2/4*42000=21000m^2s^-2

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

The points are (0,9000) and (9,21000)

The equation of the line is

v^2-9000=(21000-9000)/9t

v^2=1333.3t+9000

So,

v=sqrt((1333.3t+9000)

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

(9-0)bar v=int_0^9(sqrt(1333.3t+9000))dt

9 barv=[((1333.3t+9000)^(3/2)/(3/2*1333.3))]_0^9

=((1333.3*9+9000)^(3/2)/(2000))-((1333.3*0+9000)^(3/2)/(2000))

=21000^(3/2)/2000-9000^(3/2)/2000

=1094.7

So,

barv=1094.7/9=121.6ms^-1

The average speed is =121.6ms^-1

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