The horizontal range of a projectile with quadratic air drag is found by numerical…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

The horizontal range of a projectile with quadratic air drag is found by numerical integration with a time step Δt\Delta t, halved five times. The five computed ranges are: R1=245.6520 m,R2=245.4708 m,R3=245.3802 m,R4=245.3349 m,R5=245.3123 m.\begin{aligned}R_1 &= 245.6520\,\mathrm{m},\\R_2 &= 245.4708\,\mathrm{m},\\R_3 &= 245.3802\,\mathrm{m},\\R_4 &= 245.3349\,\mathrm{m},\\R_5 &= 245.3123\,\mathrm{m}.\end{aligned} A colleague takes the last row as the answer and reports the range as R=245.312300 mR = 245.312300\,\mathrm{m}. (a) Tabulate the successive differences R1−R2R_1-R_2, R2−R3R_2-R_3, R3−R4R_3-R_4, R4−R5R_4-R_5 and show each is close to half of the one before. (b) Say what is wrong with quoting R=245.312300 mR = 245.312300\,\mathrm{m}, and give the range with a defensible number of digits and an uncertainty, using the pattern of differences to estimate how much further the sequence would fall.

Browse the problem library