# A person in an airplane has a mass of 74 kg and 6.6 MJ of gravitational potential energy. What is the altitude of the plane?

Mar 27, 2018

The altitude is $\text{9100 m}$ or $9.1 \times {10}^{3}$ $\text{km}$.

#### Explanation:

The formula for gravitational potential energy is:

PE_"grav"=("m")("g")("h"),

where:

$\text{PE"_"grav}$ is gravitational potential energy in Joules $\left(\text{J}\right)$, $m$ is mass in kg $\left(\text{74 kg}\right)$, $g$ is acceleration due to gravity $\left(\text{9.8 m/s"^2}\right)$, and $h$ is height (altitude) in meters.

You need to convert $\text{6.6 MJ}$ to $\text{J}$.

$\text{1 J}$$=$$1 \times \text{10"^(-6)"MJ}$

$6.6 \text{MJ"xx(1"J")/(1xx10^(-6)"MJ")="6600000 J} = 6.6 \times {10}^{6}$ $\text{J}$

Solution

Rearrange the formula to isolate $h$. Plug in the known values and solve.

$h = \frac{P {E}_{\text{grav}}}{m \cdot g}$

h=(6.6xx10^6"J")/(74"kg"*"9.8m/s"^2)="9100 m"=9.1xx"10"^3" km" (rounded to two significant figures)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To see how the units cancel, refer to the following information.

$\text{1 J = 1 N"*"m}$

$1 N \cdot m = \left(1 k g \cdot \frac{m}{s} ^ 2 \cdot m\right)$

h=(6.6xx10^6(color(red)cancel(color(black)("kg")))(color(red)cancel(color(black)("m"))/color(red)cancel(color(black)("s"^2)))("m"))/(74color(red)cancel(color(black)("kg"))*9.8color(red)cancel(color(black)("m"))/color(red)cancel(color(black)("s"^2")))$=$$\text{9100 m"=9.1xx"10"^3" km}$