Solve both velocity components before integrating the path
With isotropic linear drag per unit mass gamma, horizontal and vertical equations are dvx/dt=−gamma vx and dvy/dt=−g−gamma vy. The virtual point starts at(0,5) m with velocity(4,6) m/s. Gravity is vertical but resistance acts on both components relative to the stationary medium.
Learning outcome: Compare two gamma values at the same time. Recover initial conditions and derivative identities, then distinguish vertical velocity sign from height.
Starting recap
Differentiate/integrate with explicit initial conditions; identify each force recipient and the inertial frame. Preserve force dimensions, geometric constraints and the distinction between model and solver error.
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At t=0 the formulas recover x0=0,y0=5,vx0=4,vy0=6 for every permitted gamma. Differentiate the position expressions to verify both velocity formulas.
Open the experimentPredict
With linear drag in a stationary medium, horizontal velocity here:
Experiment
Compare two gamma values at the same time. Recover initial conditions and derivative identities, then distinguish vertical velocity sign from height.
Solve both component differential equations with drag
Worked teaching model · synthetic values · Virtual point starts at(0,5) m with velocity(4,6) m/s. Acceleration=(−gamma vx,−10−gamma vy) in stationary medium. All registered evaluation times precede ground crossing; constant isotropic linear drag and g are stipulated, not a launch protocol.
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Linear-drag trajectory before ground crossing
Spatial axes use the same pixels per physical unit, preserving geometry. 1: analytic component solution (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Linear-drag trajectory before ground crossing
analytic component solution
| east position (m) | height (m) |
|---|---|
| 0 | 5 |
| 0.04984408 | 5.073986 |
| 0.0993776 | 5.145954 |
| 0.1486025 | 5.215916 |
| 0.1975207 | 5.283885 |
| 0.2461341 | 5.349872 |
| 0.2944447 | 5.41389 |
| 0.3424542 | 5.475952 |
| 0.3901646 | 5.53607 |
| 0.4375778 | 5.594255 |
| 0.4846955 | 5.650521 |
| 0.5315197 | 5.704878 |
| 0.5780521 | 5.757339 |
| 0.6242946 | 5.807915 |
| 0.670249 | 5.856619 |
| 0.7159171 | 5.903461 |
| 0.7613007 | 5.948454 |
| 0.8064014 | 5.991609 |
| 0.8512212 | 6.032938 |
| 0.8957618 | 6.072451 |
| 0.9400248 | 6.110161 |
| 0.984012 | 6.146078 |
| 1.027725 | 6.180214 |
| 1.071166 | 6.212579 |
| 1.114336 | 6.243185 |
| 1.157237 | 6.272043 |
| 1.199871 | 6.299163 |
| 1.24224 | 6.324557 |
| 1.284344 | 6.348235 |
| 1.326186 | 6.370208 |
| 1.367767 | 6.390486 |
| 1.409089 | 6.40908 |
| 1.450154 | 6.426001 |
| 1.490963 | 6.441259 |
| 1.531517 | 6.454864 |
| 1.571819 | 6.466826 |
| 1.61187 | 6.477157 |
| 1.651672 | 6.485865 |
| 1.691225 | 6.492962 |
| 1.730532 | 6.498456 |
| 1.769594 | 6.502359 |
| 1.808412 | 6.50468 |
| 1.846989 | 6.505429 |
| 1.885325 | 6.504616 |
| 1.923423 | 6.50225 |
| 1.961283 | 6.498341 |
| 1.998907 | 6.492899 |
| 2.036297 | 6.485933 |
| 2.073454 | 6.477453 |
| 2.11038 | 6.467468 |
| 2.147075 | 6.455987 |
| 2.183542 | 6.443021 |
| 2.219781 | 6.428578 |
| 2.255795 | 6.412667 |
| 2.291584 | 6.395297 |
| 2.327151 | 6.376479 |
| 2.362495 | 6.356219 |
| 2.39762 | 6.334529 |
| 2.432525 | 6.311416 |
| 2.467214 | 6.286889 |
| 2.501686 | 6.260958 |
| 2.535943 | 6.23363 |
| 2.569987 | 6.204915 |
| 2.603819 | 6.174821 |
| 2.63744 | 6.143358 |
| 2.670851 | 6.110532 |
| 2.704054 | 6.076354 |
| 2.737051 | 6.040831 |
| 2.769842 | 6.003971 |
| 2.802428 | 5.965784 |
| 2.834812 | 5.926277 |
| 2.866994 | 5.885458 |
| 2.898975 | 5.843336 |
| 2.930757 | 5.799919 |
| 2.962341 | 5.755215 |
| 2.993728 | 5.709232 |
| 3.02492 | 5.661977 |
| 3.055917 | 5.613459 |
| 3.086721 | 5.563686 |
| 3.117333 | 5.512666 |
| 3.147755 | 5.460406 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| East position | 3.147755 | m |
| Height | 5.460406 | m |
| East velocity | 2.426123 | m/s |
| North velocity | -4.230203 | m/s |
| East acceleration | -1.213061 | m/s² |
| North acceleration | -7.884899 | m/s² |
Linear resistance couples each acceleration to its own velocity component while gravity adds a vertical drive.
vx=4exp(−gamma t) and vy=(6+g/gamma)exp(−gamma t)−g/gamma. Integrating each component retains the initial5 m height. The gamma→0 limit recovers vacuum kinematics, but substituting gamma0 directly into divided expressions is not a valid numerical operation. All supported times remain above ground; no event or impact physics is added.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Linear resistance couples each acceleration to its own velocity component while gravity adds a vertical drive.
vx=4exp(−gamma t) and vy=(6+g/gamma)exp(−gamma t)−g/gamma. Integrating each component retains the initial5 m height. The gamma→0 limit recovers vacuum kinematics, but substituting gamma0 directly into divided expressions is not a valid numerical operation. All supported times remain above ground; no event or impact physics is added.
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