Question

An automobile weighing 2500 lbf increases its gravitational potential energy by a magnitude of 2.25 × 104 Btu in going from an elevation of 5183 ft in Denver to the highest elevation on Trail Ridge Road in the Rocky Mountains. What is the elevation at the high point of the road, in ft?

Answer 1

The elevation at the high point of the road is 12186.5 in ft.

Step-by-step explanation:

The automobile weight is 2500 lbf.

The automobile increases its gravitational potential energy in
`2.25 * 10^4 BTU`. It means the mobile has increased its elevation.

The initial elevation is of 5183 ft.

The first step is to convert Btu of potential energy to adequate units to work with data previously presented.

British Thermal Unit -
`1 BTU = 778.17  lbf*ft`

`2.25 * 10^4 BTU ((778.17 lbf*ft)/(1BTU) ) = 1.75 * 10^7 lbf * ft`

Now we have the gravitational potential energy in lbf*ft. Weight of the mobile is in lbf and the elevation is in ft. We can evaluate the expression for gravitational potential energy as follows:

`Ep = m*g*(h_2 - h_1)\\ W = m*g`

Where m is the mass of the automobile, g is the gravity, W is the weight of the automobile showed in the problem.

`h_2` is the final elevation and
`h_1` is the initial elevation.

Replacing W in the Ep equation

`Ep = W*(h_2 -h_1)\\(h_2 -h_1) = (Ep)/(W) \\h_2 = h_1 + (Ep)/(W)\\\\`

Finally, the next step is to replace the variables of the problem.

`h_2 = 5183 ft + (1.75 * 10^7 lbf*ft)/(2500 lbf)\\h_2 = 5183 ft + 70003.5 ft\\h_2 = 12186.5 ft`

The elevation at the high point of the road is 12186.5 in ft.