I have created a video on Youtube with the link here: https://youtu.be/BelfE5AxQwA

In which I explain the physics underlying Sayonara Umihara Kawase. The physics involved include Newton's Laws of Motion, Hooke's Law, Pendulum Motion, Simple Harmonic Motion and force vectors.

I feel it is important to explain this as it would actually help players who are stuck in the game to be able to figure out how to get past an obstacle.

When I realised I could apply physics into this game, I was able to get through stages without having to look up solutions on Youtube unless I needed to as a last resort or if I needed ideas to get a fast time for the online rankings.

Unfortunately I never received any feedback on the first video I did and this one is a revised version in which I have tried to make the explanations as I feel is clear as possible to convey how certain actions can cause Umihara Kawase to move faster by increasing her acceleration when applying certain moves.

If you have trouble understanding what is going on in the video, try watching it once more and pause where a lot of text is on screen so you don't miss the subtitles if you were paying attention to the other details.

I think that is all I needed to explain, please watch the video and enjoy.

Note: In one of the stages shown at the end of the video, the normal way to get across the large gap is the use of a trampoline that is standing almost vertically. I never could get across using that since it wouldn't work, so I managed to get past the gap without the trampoline using 2 different methods as shown because of knowing the physics in this game.

I have added a transcript here from the video. I guess this would allow some machine translation to be possible for any non-english readers to be able to understand what I am saying in the video. For those non-english readers that understand physics very well, you probably would have understood what I was showing without having to read my subtitles.

Transcript:

Understanding the traversal mechanics of Sayonara Umihara Kawase

Video created by sol-alpha a.k.a. ggx2ac

The aim of this video is to apply Newton's Laws of motion, Pendulum motion, Hooke's Law and force vectors to assist in understanding the mechanics behind the game to get through obstacles that may seem impossible.

Scenario 1: You fall into a gap that you cannot seem to jump over.

Solution: To get across, you can increase your velocity to get over the gap.

Now for the explanation of Newton's Laws and force vectors.
First, I set the fishing hook to the ground. I then press up for a moment to extend the fishing line.
I then run to the left of the fishing line and stop. As I press down, the fishing line retracts and pulls Umihara towards the right.

According to Newton's Third Law of Motion:
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

Hence, the first body is Umihara Kawase. And the second body is the fishing hook. For why the fishing hook moves Umihara I will go into more detail later. I must run right in the instant I start moving as I have to go from a state of rest to a state of momentum.

As my velocity goes to the right and acceleration is positive and increasing in that direction from pulling the fishing line, my velocity will increase.

This is due to Newton's Second Law of Motion:
The vector sum of the forces on an object is equal to the mass of that object multiplied by the
acceleration vector of the object. F = ma

I jump over the first gap. I will now explain Newton's first Law of Motion and Force Vectors

Newton's First Law of Motion:
When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

The scenario here is that it is possible to have a constant velocity which is explained by applying Force Vectors. We first draw a free-body diagram as such. There are 3 known forces acting on
Umihara Kawase. First, we look at the forces applied by gravity and the normal force. The normal force is perpendicular to the surface of contact. The force exerted by the floor prevents Umihara from falling due to gravity. In this case, the forces cancel out.

Because they are equal and opposite in magnitude and direction. The remaining known force is the applied force which is unaffected by the other force vectors due to being perpendicular to them. The force which is unknown is friction. If there was friction, as Umihara is running, the friction would have a vector going to the left which would slow down Umihara.

I do not know if a frictional force is applied on flat surfaces like this one in the game. Umihara can slow down when running up ramps, so frictional forces may apply there. To conclude, this explains that Umihara is currently running at a constant velocity. It will not change unless a force
is exerted onto her. I jump once more. And make it across the gap.

Traversal via Hooke's Law and Pendulum Motion

What you are seeing right now is an application of Hooke's Law it is characterised by the force
needed to extend or compress a spring by some distance. The formula is F = -kx, where 'F' is the force of the mass attached to the free end of the spring, 'k' is the spring constant where a
small positive value means it is elastic and a large 'k' value means the spring is rigid.

The 'x' value is the displacement of the spring, usually written as x - x_0 where x is the spring's
'relaxed' state/value and x_0 is the current position of the spring. When you press down, Umihara compresses the 'spring' exerting a force that is proportional to the current length of the
spring. The longer the length, the greater the force that needs to be applied. This should look
familiar.

Because Hooke's law is being applied here, because the fishing hook is characterised as a spring, the force exerted by Umihara changes the displacement of the spring by it's elasticity.
Moving on.

Next, we have pendulum motion. It is difficult to simulate a pendulum because the oscillations (going up and down) from the 'spring' affect the motion. If we press left/right to swing side by side, we can reduce the oscillations and simulate "Simple Harmonic Motion".

Simple Harmonic Motion is a periodic motion where the restoring force is proportional to the
displacement and acts in the direction opposite of that displacement. In this case, Umihara is
the restoring force as she accelerates back to the centre position where the displacement is zero. Simple Harmonic Motion is a motion that obeys Hooke's Law as it occured earlier where oscillations were going up and down that a spring also uses Simple Harmonic Motion.

Going back to pendulum motion, the forces acting on a pendulum are the tensional forces which are in the direction along the fishing line, towards the fishing hook at all times. Forces from gravity and the mass of the object act directly downwards, at all times. Because the forces aren't equal in magnitude and direction they will not cancel each other out and cause Umihara to accelerate increasingly towards the centre and then accelerate in the opposite direction when moving to the other end from the centre.

To put this another way in terms of energy Umihara is at the left end of the pendulum, her velocity is at zero due to the displacement from the centre of the pendulum. (Maximum velocity occurs at the centre)
Acceleration will take effect and increase in direction towards the centre. Her potential (stored) energy is converted to kinetic energy which is the energy it has due to motion, which is
affected by mass and speed.

As Umihara goes past the centre, her kinetic energy will will decrease and potential energy will increase. This is due to acceleration going towards the opposite direction and slowing her down.
Keep in mind this diagram utilises Conservation of momentum. As long as no external forces act upon Umihara, Momentum will be conserved. Also, the symbols and energy states are
flipped in pendulum motion for going in the opposite direction.

Now to apply both Hooke's Law and Pendulum motion. Once I have gone past the centre, I
pressed down and compressed the 'spring', my acceleration to the right wasn't large and was starting to slow me down so pressing down accelerated me upwards, this would not be cancelled
by the accelerating force of the pendulum motion.

As shown here, I pressed down to compress the spring and then let go of the hook before I accelerated past the centre.This combined the acceleration going up and to the right giving the result shown.

Here's another example for your convenience. ...and another.

To summarise: Utilising forces from all these physical laws will increase your velocity and allow you get get past difficult obstacles.

Now, some actual examples without explanation, to see in action all the things you have learned
becoming applied.

Stage 17

Stage 17

Stage 7

Method 1

Method 2

Method 1

Method 2

All physics related information have been sourced from Wikipedia

The game played here was Sayonara Umihara Kawase and can be found on Steam for PC (and
Nintendo 3DS and Playstation Vita)