I eat a 500 Calorie lunch and go hiking up a mountain afterwards. Assuming that chemical energy is converted to gravitational potential energy with 50% efficiency, approximately how many miles up the mountain can I go before I need my dinner?
0.5 miles
1 mile
2 miles
3 miles
4 miles
To calculate the distance you can hike up the mountain before needing dinner, we need to convert the 500 calories from your lunch into gravitational potential energy.
First, we convert the calories to joules using the conversion factor: 1 calorie = 4.184 joules. So, 500 calories = 500 * 4.184 = 2092 joules.
Next, we need to consider the conversion efficiency. Given that you convert chemical energy to gravitational potential energy with 50% efficiency, we need to calculate how much energy is actually converted. Multiply the energy by 50% efficiency: 2092 joules * 0.50 = 1046 joules.
Now, to determine the distance you can hike, we can use the equation for gravitational potential energy: potential energy = mass * gravity * height.
Assuming your mass is constant and gravity remains the same, the potential energy is directly proportional to the height. So, we can create a proportion:
(500 calories) / (4 miles) = (1046 joules) / (x miles)
Cross-multiplying, we have:
(500 * x) = (4 * 1046)
500x = 4184
x = 4184 / 500
x = 8.368
Therefore, you can hike approximately 8.368 miles up the mountain before needing dinner. Since this value is between 4 miles and 2 miles, the closest answer choice would be 4 miles.