Newton's Laws

A force model represents each interaction between two objects by a force vector whose size comes from a measured force law, and it applies in an inertial frame — one in which an object with no net interaction keeps a constant velocity. Use the supplied topic model Y_out = 2*u^2*f*(1-f)/g, where Y_out is HEIGHT ABOVE DECK, BOTH (m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; g = 9.81 m s⁻². f is Moment in the flight t/T, dimensionless. Keep vc (Flatcar speed V) = 3 m s⁻¹; u (Throw speed u, relative to the flatcar) = 5 m s⁻¹ fixed. For this exercise use only 0.52625 ≤ f ≤ 0.89375 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 = 0.95457349 m, with a bounded reading uncertainty ±0.047419 m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00003675, 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.

Newton's first law: in an inertial frame, a body on which the net external force is zero keeps a constant velocity, staying at rest if it was at rest, and any body moving with constant velocity has zero net force on it. Use the supplied topic model Y_out = f1-f2, where Y_out is NET FORCE ΣF (N). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. f1 is Rightward pull F₁, measured in N. Keep f2 (Leftward pull F₂) = 6 N; v0 (Starting velocity v₀) = 1.5 m s⁻¹ fixed. For this exercise use only 2.85 N ≤ f1 ≤ 9.15 N 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 = 0.693 N, with a bounded reading uncertainty ±0.36855 N. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f1 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00063 N, 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.

Newton's second law says that, in an inertial frame, the vector sum of the external forces on a body equals the product of its inertial mass and its acceleration, so a = ΣF/m and the acceleration points along the net force. Use the supplied topic model Y_out = hypot(5+f2*cosd(th), f2*sind(th))/m, where Y_out is ACCELERATION |a| (m s⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); sind/cosd/tand use degrees. m is Puck mass m, measured in kg. Keep f2 (Size of second force F₂) = 4 N; th (Angle θ between F₁ and F₂) = 60 ° fixed. For this exercise use only 1.2125 kg ≤ m ≤ 2.7875 kg 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 = 3.593811 m s⁻², with a bounded reading uncertainty ±0.11879 m s⁻². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 kg, 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.

Newton's third law: when object 1 exerts a force on object 2, object 2 exerts a force on object 1 of equal magnitude and opposite direction, of the same type and at the same instant, so F(2 on 1) = −F(1 on 2). Use the supplied topic model Y_out = m*(g+a), where Y_out is HAND ON BOOK = BOOK ON HAND (N). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; g = 9.81 m s⁻². m is Book mass m, measured in kg. Keep a (Acceleration a (up is +)) = 3 m s⁻² fixed. For this exercise use only 1.2125 kg ≤ m ≤ 2.7875 kg 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 = 27.839332 N, with a bounded reading uncertainty ±1.1803 N. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 kg, 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.