# An object has a mass of 4 kg. The object's kinetic energy uniformly changes from 96 KJ to  280 KJ over t in [0, 12 s]. What is the average speed of the object?

Feb 16, 2017

The average speed is $= 296.65 m {s}^{-} 1$

#### Explanation:

The initial kinetic energy is $= \frac{1}{2} m {u}^{2}$

So,

$\frac{1}{2} m {u}^{2} = 96000$

${u}^{2} = 2 \cdot \frac{96000}{4} = 48000$

$v = \sqrt{48000} = 219.1 m {s}^{-} 1$

The final kinetic energy is $\frac{1}{2} m {v}^{2}$

So,

$\frac{1}{2} m {v}^{2} = 280000$

${v}^{2} = 2 \cdot \frac{280000}{4} = 140000$

$v = \sqrt{140000} = 374.2 m {s}^{-} 1$

On a graph velocity v/s time, we have a straight line $D C$

$A D = u$

$B C = v$

$A B = t$

Area of $A B C D = \frac{A D + B C}{2} \cdot t$

The average speed $= \overline{v}$

$\overline{v} \cdot t = \frac{A D + B C}{2} \cdot t$

so,

$\overline{v} = \frac{u + v}{2}$

$= \frac{219.1 + 374.2}{2}$

$= 296.65 m {s}^{-} 1$