Kinetic-theory model of pressure

The kinetic-theory model of pressure treats gas pressure as the time-averaged force per unit area that molecules deliver by bouncing elastically off the container walls, which makes p = ⅓ n m ⟨v²⟩ for number density n, molecular mass m and mean-square speed ⟨v²⟩. Use the supplied topic model Y_out = vrms*cosd(theta), where Y_out is NORMAL COMPONENT v cosθ (m s⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Angle to the wall normal θ, measured in °. Keep vrms (Rms molecular speed v_rms) = 500 m s⁻¹; nd (Number density n) = 2.5 ×10²⁵ m⁻³ fixed. For this exercise use only 15.3125 ° ≤ theta ≤ 54.6875 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at theta = 23.975 ° and 29.4875 °. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δtheta across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

The kinetic-theory model of pressure treats gas pressure as the time-averaged force per unit area that molecules deliver by bouncing elastically off the container walls, which makes p = ⅓ n m ⟨v²⟩ for number density n, molecular mass m and mean-square speed ⟨v²⟩. Use the supplied topic model Y_out = vrms*cosd(theta), where Y_out is NORMAL COMPONENT v cosθ (m s⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Angle to the wall normal θ, measured in °. Keep vrms (Rms molecular speed v_rms) = 500 m s⁻¹; nd (Number density n) = 2.5 ×10²⁵ m⁻³ fixed. For this exercise use only 15.3125 ° ≤ theta ≤ 54.6875 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 386.74732 m s⁻¹, with a bounded reading uncertainty ±14.325 m s⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0039375 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.