Imagine that your car can travel 25 mi on 1 gal of gas when traveling on a level road at 55mi/h. Estimate your gas mileage while traveling at 55 mi/h up an incline that gains 3m in elevation for every 100 m one goes forward along the incline. Assume that your car has a mass of 1500 kg.
At a playground, a 19 kg child plays on a slide that drops through a height of 2.3. The child starts at rest at the top of the slide. On the way down, the slide does a nonconervative work of -361 J on the child. What is the child's speed at the bottom of the slide?
To estimate the gas mileage while traveling at 55 mi/h up an incline, we need to consider the additional energy required to overcome the gravitational potential energy of the incline.
First, let's convert the given values for fuel efficiency and speed into SI units for consistency:
1 mile = 1609.34 meters
1 gallon = 3.78541 liters
The car's fuel efficiency on a level road can be calculated as follows:
Gas mileage on level road = Distance traveled / Fuel consumption
= 25 miles / 1 gallon
= 25 * 1609.34 meters / 3.78541 liters
≈ 10.56 meters/liter
Now, let's calculate the total energy required to overcome the gravitational potential energy for each 100 meters traveled uphill:
Energy required per 100 meters = Mass * g * Height
= 1500 kg * 9.8 m/s^2 * (3 m / 100 m)
≈ 441 Joules
Next, we need to calculate the additional fuel consumption required to generate this energy. The energy provided by the fuel can be calculated using the energy density of gasoline:
Energy provided per liter of gasoline = Energy density of gasoline * Volume
= 32 MJ/liter
= 32 * 10^6 Joules/liter
Additional fuel consumption per 100 meters uphill = Energy required per 100 meters / Energy provided per liter of gasoline
= 441 Joules / (32 * 10^6 Joules/liter)
≈ 1.378 * 10^-5 liters
Now, let's calculate the gas mileage (meters per liter) while traveling uphill:
Gas mileage uphill = Distance traveled / (Fuel consumption on level road + Additional fuel consumption per 100 meters uphill)
= 10.56 meters/liter / (10.56 meters/liter + 1.378 * 10^-5 liters/meter)
≈ 10.56 meters/liter
Therefore, the estimated gas mileage while traveling at 55 mi/h up an incline that gains 3m in elevation for every 100 m forward along the incline is approximately 10.56 meters per liter, which is the same as the gas mileage on a level road.