Why is it harder to walk up the hill to class?

 Why is it harder to walk up the hill to class?

Like the majority of the Colgate population, I find it harder to walk up the hill to get to class than to walk back down when going home. Walking up the hill to get to class when running late is not the most enjoyable part of my day, thus the process of leaving my dorm, walking up the never-ending series of stairs that exist on Colgate’s campus to get to my class all the way up the hill is therefore quite the feat.

But as a physics student, I wondered whether there was perhaps a mathematical and physics-related explanation for why the journey of walking up to class feels as if it is harder and longer than the walk back to my dorm.

As I live in 113 Broad Street, part of my walk to class includes walking up the hill. Assuming that my path is “ideal” and represents a straight walk up a slope, an analysis of the forces on me can be done.

When I am walking up the hill, there are two forces acting on me - the force of gravity and the force of friction. Assuming that I realize I am desperately late for class, I gradually increase the pace at which I am walking, thus my acceleration is greater than zero. Thus, as I exert acceleration  in the +x direction, both gravity and friction exert force in the -x direction, resulting in an overall decrease of acceleration than my own acceleration.

When I am walking down the hill, there are still two forces acting on me - the force of gravity and the force of friction. Assuming I am excited to race back to my room, I gradually accelerate the pace of my walk. This time, as I am walking in the -x direction, gravity is acting upon me in the -x direction as well (while friction acts in the +x direction). Thus, the magnitude of my overall acceleration is increased when walking down the hill, as I no longer have two forces acting in the opposite direction of my velocity (and acceleration), and instead only have one. 

Thus, the reason why it seems like a more difficult journey to walk up the hill to get to class is because there are forces acting in the opposite direction than the direction I am moving in, thus slowing me down and requiring a longer walking time!