# An object has a mass of 9 kg. The object's kinetic energy uniformly changes from 135 KJ to  45 KJ over t in [0, 4 s]. What is the average speed of the object?

Average Speed $= 136.6025404 \text{ }$m/sec

#### Explanation:

The given data:
Mass $m = 9 \text{ }$kg
Kinetic Energy initial $K {E}_{i} = 135000 \text{ }$joule
Kinetic Energy final $K {E}_{f} = 45000 \text{ }$joule

initial time ${t}_{0} = 0 \text{ }$sec
final time ${t}_{1} = 4 \text{ }$sec

Compute for initial velocity ${v}_{i}$

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

$135000 = \frac{1}{2} \left(9\right) {v}_{i}^{2}$

${v}_{i}^{2} = \frac{2 \left(135000\right)}{9}$

${v}_{i} = \sqrt{30000}$

${v}_{i} = 173.2050808 \text{ }$m/sec

Compute for final velocity ${v}_{f}$

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

$45000 = \frac{1}{2} \left(9\right) {v}_{f}^{2}$

${v}_{f}^{2} = \frac{2 \left(45000\right)}{9}$

${v}_{f} = \sqrt{10000}$

${v}_{f} = 100 \text{ }$m/sec

Solve for the total distance traveled by the object using the following:

${v}_{f}^{2} = {v}_{i}^{2} + 2 a \cdot s \text{ }$first equation
and
${v}_{f} = {v}_{i} + a \cdot t \text{ }$second equation

${v}_{f}^{2} - {v}_{i}^{2} = 2 a \cdot s \text{ }$from the first equation

$\textcolor{red}{\left({v}_{f} + {v}_{i}\right) \left({v}_{f} - {v}_{i}\right) = 2 \cdot a \cdot s}$

$\textcolor{b l u e}{\left({v}_{f} - {v}_{i}\right) = a \cdot t \text{ }}$from the second equation

Divide the first equation by the second equation

$\frac{\textcolor{red}{\left({v}_{f} + {v}_{i}\right) \cancel{{v}_{f} - {v}_{i}}}}{\textcolor{b l u e}{\cancel{{v}_{f} - {v}_{i}}}} = \frac{\textcolor{red}{2 \cdot \cancel{a} \cdot s}}{\textcolor{b l u e}{\cancel{a} \cdot t}}$

${v}_{f} + {v}_{i} = \frac{2 s}{t}$

$s = \frac{1}{2} \cdot t \left({v}_{f} + {v}_{i}\right)$

$s = \frac{1}{2} \cdot \left(4\right) \left(100 + 173.2050808\right)$

$s = 546.4101616 \text{ }$m

Average speed $= \left(\text{distance traveled")/("elapsed time}\right)$

Average speed $= \frac{546.4101616}{4}$

Average speed $= 136.6025404 \text{ }$m/sec

God bless....I hope the explanation is useful.