The range of a projectile launched with speed at angle above the horizontal (in…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians

Problem

The range of a projectile launched with speed vv at angle θ\theta above the horizontal (in radians) on level ground, with air resistance ignored, is R(v,θ)=v2sin⁡(2θ)/gR(v,\theta) = v^2\sin(2\theta)/g, with g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}}. Someone estimates how much the range changes when v=20.0 m s−1v = 20.0\,\mathrm{m\,s^{-1}} and θ=0.50 rad\theta = 0.50\,\mathrm{rad} are changed by Δv=+0.30 m s−1\Delta v = +0.30\,\mathrm{m\,s^{-1}} and Δθ=+0.020 rad\Delta\theta = +0.020\,\mathrm{rad}, writing: (∂R∂v)θ=2vsin⁡(2θ)g,(∂R∂θ)v=v2cos⁡(2θ)g\left(\frac{\partial R}{\partial v}\right)_\theta = \frac{2v\sin(2\theta)}{g}, \qquad \left(\frac{\partial R}{\partial \theta}\right)_v = \frac{v^2\cos(2\theta)}{g} ΔR≈(∂R∂v)θΔv+(∂R∂θ)vΔθ≈1.47 m\Delta R \approx \left(\frac{\partial R}{\partial v}\right)_\theta \Delta v + \left(\frac{\partial R}{\partial \theta}\right)_v \Delta\theta \approx 1.47\,\mathrm{m} ((a)) Find the error in this working. ((b)) Give the corrected estimate of ΔR\Delta R.

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