Yesterday I climbed Giant Mountain, one of the 46 Adirondack High Peaks. With a summit elevation of 4,627 feet (1,410 m) Giant is the 12th tallest of the high peaks and with an elevation change of 3000 ft in 3 miles it's also on of the steepest. The journey began at the car near the trail head where I was deciding on footwear. The 2 options were hiking boots (0.92 kg a pair) of Nike frees (.42 kg a pair). The boots would be heavier and require more work to ascend the mountain, but would provide better traction and keep my feet dry. The frees would require less energy but likely slip on everything, provide less support and get my feet drenched within minutes. I chose the boots, so how much more work did I do climbing the mountain? The ideal approach to figuring this out would be to multiply the number of steps that I took while ascending and descending the mountain by the average distance that I lifted my feet with each step; and then multiply that by the force I exerted against the weight of my boots/shoes (work=force*displacement). However I didn't count my steps because counting for 5 hours would have driven me insane and the vertical distance that I lifted my feet varried widely on the diffenrt sorts of terain I encountered. So I'll just use the vertical displacement up the mountain as my displacement. The difference im energy expendature can be found by multiplying the difference in weight of the shoes by the displacement up the mountain.
Difference in weight=(.92kg-.42kg)(9.8m/s2)=4.9N
Difference in work=(4.9N)(914m)=4497.6J
So by choosing the boots I expended about an extra 4500 Joules (about 1070 calories) of energy (but I estimate that in reality It was probably closer to double that). However as we climbed further the trail became covered in snow and ice, making it incredibly wet and slippery, so without the boots I likely would have fallen off the the mountain and gotten frostbite on my feet. In the end think 4500J is a fair tradeoff for not dying.