Chapter Text
Success! The flute collides with the arrow and they stick together for a brief second before the flute breaks into a hundred tiny splinters.
Calculate the energy lost as heat in the collision.
Calculate the deflection of the arrow.
a, b) We can solve these two subproblems in one go using conservation of momentum.
Pros: Conservation of momentum questions are mathematically very simple.
Cons: This is a conservation of momentum question in three dimensions. Three .
To avoid confusion, let’s impose a coordinate system. Let z be the vertical axis, let x be the axis parallel to the wall, and let y be the axis perpendicular to the wall. To make our lives easier, let’s say the direction of the flute and the direction of the arrow are both positive.
Let’s look at the flute first. From the previous section, its x-velocity is 34.78 m/s . Its y-velocity is zero (thank god). Its z-velocity is v_0-g t, where v_0 is 23.98 m/s, g is 10 m/s^2, and t is 0.23 s, so around 21.68 m/s.
For the arrow, we assume the speed stays constant at around 84.09 m/s . We can split this into components via the 7-24-25 triangle to get its y-velocity as 80.73 m/s and its z-velocity as 23.55 m/s.
Now for the fun part: set up and solve the dang equations.
X direction: (34.78 m/s) (0.2 kg) = (0.2+0.15 kg) [final x velocity].
Y direction: (80.73 m/s) (0.15 kg) = (0.2+0.15 kg) [final y velocity].
Z direction: (21.68 m/s) (0.2 kg) + (23.55 m/s) (0.15 kg) = (0.2+0.15 kg) [final z velocity].
Solving, we achieve a final velocity vector of (19.87, 34.59, 22.48) m/s.
From these values, we calculate the heat, which is all kinetic energy loss. Note that the v^2 part of kinetic energy (½ m v^2) can be calculated by summing the squares of the components.
I won’t trifle you with the (rather ugly) equations, but the final heat loss approximates 331.39 J.
Finally, we compare the final and initial velocities of the arrow to calculate its deflection. Using the dot product, we calculate the angle of deflection to be around 56.35 degrees.
Now, this leaves only the problem of a literal entire room filled with armed cultivators, the vast majority of which are out for Wei WuXian’s blood.
To this, I call upon the wisdom of countless generations of textbook writers past:
The answer is trivial and left as an exercise to the reader.
Q. E. D.