# 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, 6 s]. What is the average speed of the object?

Apr 15, 2016

Find the average kinetic energy and convert it to a speed, given the mass. The average speed is $3.08$ $m {s}^{-} 1$.

#### Explanation:

Let me outline how I approached this problem, including what turned out to be a dead end.

Kinetic energy is given by:

${E}_{k} = \frac{1}{2} m {v}^{2}$.

Rearranging this to make $v$ the subject gives:

$v = \sqrt{\frac{2 {E}_{k}}{m}}$.

Using this, the initial velocity of the object is $\sqrt{14} \approx 3.74$ $m {s}^{-} 1$ and its final velocity is $\sqrt{5} \approx 2.24$ $m {s}^{-} 1$.

If the speed change uniformly we could simply add these two speeds and divide by 2 to find the average, but the tricky bit here is that the question says the kinetic energy changes uniformly... and kinetic energy is proportional to the square of the speed.

Given that, we need to approach the problem from a different direction. The average kinetic energy will be given by:

${E}_{k} = \frac{42 + 15}{2} = 28.5$ $k J$

Using the formula above for the speed gives:

$v = \sqrt{\frac{2 {E}_{k}}{m}} = \sqrt{\frac{2 \cdot 28.5}{6}} = \sqrt{9.5} \approx 3.08$ $m {s}^{-} 1$.