General Mechanics
Part 2: Jump Height
When it comes to our player, he has a limited jump height. However, since we have calculated gravity to be ≈ 11.59m/s², we can figure out what speed would be required for them to jump up to a particular height.
Using the kinematic equation v² = v_0² + 2*g*Δy, we can find that 0 = v_0² - 2*11.59m/s²*2m => v_0 = sqrt(46.36m²/s²) = 6.809m/s when not using any equipment.
Using the blade runners, however, the max height becomes 0 = v_0² - 2*11.59m/s²*4m => v_0 = sqrt(92.72m²/s²) = 9.629m/s on lift-off.
To calculate the force required to get off the ground, we would require the mass of the player. However, there isn't any good way of measuring the mass of the player. We will, instead, make the mass be the value M.
From kinematics, F = m * Δv/Δt, so the force required to lift off the ground would be:
F = m * (6.809m/s)/(1s/60) = 408.54*M m/s² without extra technology,
F = m * (9.629m/s)/(1s/60) = 577.74*M m/s² with the "blade runners."
F = (408.54+11.59)*M m/s² = 420.13*M m/s² without extra technology + gravity
F = (577.74+11.59)*M m/s² = 589.33*M m/s² with the "blade runners" + gravity
However, there is a bit of weird physics at play when the player is actually sliding while they do the jump. Their jump height, for whatever reason, is increased!
Doing the slide jump adds an extra 50% jump height increase.
Using the equations above, we get a initial velocity in the vertical direction off the ground of 8.34m/s without extra tech and 11.793m/s with the "blade runners."
The force off the ground (once again assuming the force is applied over 1 frame in game being 0.0167s) will be the following:
F = m * (8.34m/s)/(1s/60) =*M m/s² without extra technology,
F = m * (11.793m/s)/(1s/60) =*M m/s² with the "blade runners."
F = (8.34+11.59)*M m/s² =*M m/s² without extra technology + gravity
F = (11.793+11.59)*M m/s² =*M m/s² with the "blade runners" + gravity
Once again, we will have the player's mass be the value M.
Normally, in the real world, the horizontal velocity of an object doesn't (classically) impact the vertical velocity in projectile motion.
Either the slide provides an extra boost like a power jump from Mario 2 or the game physics engine take into account horizonal velocity when calculating jump physics.