When a child swings on a swing, have you ever thought how it is possible without touching the ground. There is nothing to push off against.. a naive application of conservation of momentum might indicate that it is not even possible to start swinging!
So how does it work? The catch is twofold. The first point is gravity, of course. Conservation of momentum applies in the absence of external forces. The second point is the person on the swing does not act as a point mass. Instead the simplest model one can use is a two point mass model, where the legs are one mass element and the torso is the other mass element.
When at the apex or top of the backward motion swing, the person would kick out their legs. This launches their torso and the swing backwards while projecting the leg forwards/downwards. Here is the crux.. the torso then does not decelerate by pulling back on the legs and hence conserving momentum. Instead, gravity decelerates the torso, storing up potential energy. This adds amplitude to the swing as desired. When the swing direction reverses then at the apex, the body follows the legs finally.
This bring us to a limitation of this technique. The person cannot detach his legs of course. So there is no benefit to making the kick too aggressive. The optimal velocity of the kick would be such that the knees reach full extension in the duration that gravity decelerates the torso to reverse swing direction. Any faster and the legs get pulled back by the body trying to not break apart and the additional force is for naught.
The final point is that all this is applicable only when starting with some small non-zero swing amplitude. Thus, it’s really difficult (not sure if it is impossible) to start swinging when perfectly stationary as an initial condition. At that stage, the person might just touch and use the ground, to get started.
Cars and Physics 