Question

An object has a mass of `4 kg`. The object's kinetic energy uniformly changes from `16 KJ` to `96KJ` over `t in [0, 6 s]`. What is the average speed of the object?

Answer 1

163.345

Explanation:

Here, let's assume that uniform change means that the graph of Kinetic energy vs. time is a straight line.

Let y coordinate be kinetic energy, and x coordinate be time. As at x=0, y=16000, so y-intercept of line is 16000.

Slope of the line is `(96000-16000)/(6-0)=40000/3`

Equation of line is
`y = (40000/3)x + 16000`

`K.E. = (40000/3)t + 16000`

`1/2 mv^2 = (40000/3)t + 16000`
Putting value of m as 4 and simplifying:

`v^2 = (20000/3)t + 8000`
`v = sqrt((20000/3)t + 8000)`

Putting `v=(ds)/dt`, taking dt to R.H.S and integrating from t=0 to t=6, we get

`s = int_0^6(sqrt((20000/3)t + 8000))dt`

Upon integrating (which I leave up to you), `s~~980.07`

We know that average speed = total distance / total time.

So Avg. Speed `~~980.07/6`

Avg. Speed `~~163.345 m/s`