# An object has a mass of 6 kg. The object's kinetic energy uniformly changes from 42 KJ to  15 KJ over t in [0, 8 s]. What is the average speed of the object?

Apr 6, 2017

The average speed is $= 96.5 m {s}^{-} 1$

#### Explanation:

The kinetic energy is

$K E = \frac{1}{2} m {v}^{2}$

mass is $= 6 k g$

The initial velocity is $= {u}_{1}$

$\frac{1}{2} m {u}_{1}^{2} = 42000 J$

The final velocity is $= {u}_{2}$

$\frac{1}{2} m {u}_{2}^{2} = 15000 J$

Therefore,

${u}_{1}^{2} = \frac{2}{6} \cdot 42000 = 14000 {m}^{2} {s}^{-} 2$

and,

${u}_{2}^{2} = \frac{2}{6} \cdot 15000 = 5000 {m}^{2} {s}^{-} 2$

The graph of ${v}^{2} = f \left(t\right)$ is a straight line

The points are $\left(0 , 14000\right)$ and $\left(8 , 5000\right)$

The equation of the line is

${v}^{2} - 14000 = \frac{5000 - 14000}{8} t$

${v}^{2} = - 1125 t + 14000$

So,

v=sqrt((-1125t+14000)

We need to calculate the average value of $v$ over $t \in \left[0 , 8\right]$

$\left(8 - 0\right) \overline{v} = {\int}_{0}^{8} \sqrt{\left(- 1125 t + 14000\right)} \mathrm{dt}$

8 barv=[((-1125t+14000)^(3/2)/(-3/2*1125)]_0^8

$= \left({\left(- 1125 \cdot 8 + 14000\right)}^{\frac{3}{2}} / \left(- 1687.5\right)\right) - \left({\left(- 1125 \cdot 0 + 14000\right)}^{\frac{3}{2}} / \left(- 1687.5\right)\right)$

$= - {5000}^{\frac{3}{2}} / 1685.5 + {14000}^{\frac{3}{2}} / 1687.5$

$= 772.1$

So,

$\overline{v} = \frac{772.1}{8} = 96.5 m {s}^{-} 1$