How to calculate when wheels are better than legs? We all know that it uses less energy to go by wheeled transport than walking if the terrain is smooth and there are not too many hills. We also all know that when going up steep hills we get off whatever wheeled transport we are using and walk.  To make things concrete, let us consider a child's scooter.

How can one calculate the steepness of the hill at which point you would use more energy to scoot than to walk?
 A: This is indeed getting into biomechanics. But there are a few issues here in addition to the ones mentioned so far.
As mentioned, the scooter itself has mass. Hence, moving the scooter up the hill requires work. This is an advantage to the walker.
On a level surface, walking is not as efficient a mode of transportation as wheels. This is fairly obvious -- you can ride a bike much faster than you can walk (or, for that matter, than you can run). One of the main culprits in walking is that in order to walk, you must repetitively raise your center of gravity (COG) up and down. It takes work to raise up your COG. When you then lower your COG, however, only a small part of that energy goes into forward motion; most gets wasted as heat when your feet strike the pavement or when your cartilage, ligaments and tendons absorb the impact.
This wasted effort in walking is relatively independent of what the incline is. So when you walk up a steep hill, you're still wasting some energy, but you're expending a huge amount of energy just moving your body mass up the hill. Thus, your efficiency (work that you want to do divided by total work) is better as you walk up a steeper and steeper hill.
At some point of steepness, extra work of moving the scooter's mass up the hill outweighs the inefficiency of the walking motion.
Another issue that prevents walking from working well at high velocities (but does not hamper you walking up a hill) is that walking is essentially a repetition of circular motion around an axis. Your left foot plants on the ground and is an axis for rotation until the right foot plants. Then your body rotates around your right foot. If you try to walk too fast, then the equations of centripetal force tell you that the forces involved in walking become high enough that you leave the ground and you are suddenly running instead of walking. See, e.g., College Physics AP Edition, Randall Knight, Chapter 6 for a good explanation of this. The bottom line, though, is that again walking works better at low speed than at high speed.
In general, wheeled transport is very efficient. Rolling at a constant speed on a level surface requires only overcoming rolling resistance and air drag, neither of which are large factors (in a relative sense). Human motion via muscles, on the other hand, is not nearly so efficient. In physics, work = force * distance. In human physiology, muscles expend chemical energy and produce lactate (which makes you feel tired) even when they are performing a static contraction (i.e., you tense your muscles but don't move) or an eccentric contraction (e.g., you lower a weight to the ground). Walking requires what a biomechanics person would call a "kinetic chain"; i.e., cooperation of muscles from your hips to your toes to make your entire leg do what you want it to. Even if, e.g., your knee muscles do nothing but hold your knee joint rigid during part of walking and (according to physics) do no work, they still burn chemical energy.
So on a flat surface, wheeled transport is dominated by the super-efficient gliding. Yes, you must move your muscles to impart enough energy to get your scooter or bicycle up to speed and overcome friction. But these are not a big deal, and mostly you're gliding. Up a hill, though, your scooter or bicycle must itself go up the hill, requiring substantial power that soon becomes bigger than the friction from rolling resistance and air drag. Even if bicycle is very lightweight, you must still get your body up the hill, which requires way more power than frictional forces do. You can try to get around this by going slowly (e.g., using the granny gears on a bicycle) -- but soon you go so slowly that your bicycle wobbles and your scooter risks going backwards. Conclusion: wheels are great on the flats, but not so great uphill.
Walking is the reverse. On a level surface, walking is as mentioned quite inefficient. On a hill, though, you expend so much energy just overcoming gravity that the inefficiencies of human locomotion aren't (relatively speaking) so bad. Another interesting point is that when a hill gets steep enough, most people change the way they walk. On a level surface, their heel strikes the ground before their sole or toes, resulting in contact forces that oppose their motion and hence waste energy as heat. When the hill gets steep, they strike with their sole or even toes, resulting in contact forces with a component that gets you up the hill.
For a nice biomechanical analysis of walking, see Chapter 19 of "Kinesiology, Scientific Basis of Human Motion" by Hamilton, Weimar and Luttgens. It was my undergrad kinesiology textbook way back when it was only on the 5th edition, and now they're on the 12th edition. So I won't say how old I am :-)
A: The following might be how you approach the solution to this - but you will need to consider details of biomechanics for a "real" answer:
When you are on a scooter, you carry a small amount of extra weight. This means you always "do more work" (in the physics sense) to go uphill. But the work consists of two components:
1) work to go along horizontally
2) work against gravity
Scooters and such make (1) easier but they do nothing for (2). As such they are useful when a simple push makes you roll for an appreciable distance. As the hill gets steeper you will come to a halt more quickly, and in order to prevent yourself from rolling backwards you need to quickly pull your leg forward for the next push. When the speed required to move your leg forward is greater than the free pendulum time (what would happen if you let your leg swing forward with the force of gravity) you end up doing extra work.
So the things to consider:


*

*Weight of scooter

*Velocity at end of push

*Time it takes to lose this velocity given the slope

*Time it takes for leg to swing forward naturally (see this earlier question)


When the time in 4 exceeds the time in 3, you are better off walking. I will write down equations, but they are very approximate...
If the slope has a slope G (expressed as the sine of the angle), then the apparent force slowing down the scooter is $f_a=F_g\cdot G=mgG$. A scooter with initial velocity $v$ will slow down in a time $t=mv/f_a=\frac{v}{gG}$.
Assuming that each push of the leg gets you to walking speed, (5km/h) and that your leg takes 0.5 seconds to swing back without extra effort, we can now solve for G:
$$G=\frac{v}{gt}\\
=\frac{1.4}{5}\\=
0.28$$
This corresponds to a 1:4 slope - about 16 degrees - as the limiting case. In reality pushing yourself to walking speed in each push up a slope that steep might take considerable effort so this is an upper bound.
A: In a "purely physical" calculation, as long as the scooter did not roll backwards, then you would do the same amount of work getting up the hill with the scooter whether you walked, ran or scooted.
