Predict → set conditions → observe → record → change one variable. Loading an example clears records and creates a reproducible paused state.
01 · Independent components
Gravity changes only vertical velocity. A dropped and a horizontally launched object from the same height land together in vacuum.
x=v_0\cos\theta\,t,\qquad y=h+v_0\sin\theta\,t-\frac{1}{2}gt^2
Try itLoad h=20 m and a horizontal speed of 10 m/s, seek to the end and record. Change only the speed to 20 m/s and record again.
Prediction. Both flight times are 2.00 s; the ranges are 20.00 m and 40.00 m.
Compare and explain. Why does doubling horizontal speed leave fall time unchanged? Explain using acceleration.
Model limit. With drag, vertical resistance depends on total speed, so this comparison changes.
02 · Motion at the apex
At the apex the vertical velocity is zero; total velocity and acceleration need not vanish.
t_\mathrm{top}=\frac{v_0\sin\theta}{g},\qquad H=h+\frac{v_0^2\sin^2\theta}{2g}
Try itLoad the apex for 20 m/s at 30° with g=10. Show velocity, acceleration and components; compare with a 90° launch.
Prediction. At t=1.00 s, y=5.00 m, vx=17.32 m/s and vy=0.
Compare and explain. Why does a vertically launched object fall again even though its apex speed is zero?
Model limit. A tiny horizontal component at 90° may be floating-point residue.
03 · Complementary angles
At equal launch and landing heights, 30° and 60° give the same range but different flight times and heights.
R=\frac{v_0^2\sin 2\theta}{g}
Try itRecord the 30° result, change only the angle to 60°, seek to the end and record. Compare paths and flight times.
Prediction. Both ranges are 34.64 m. Flight times are 2.00 and 3.46 s; peak heights are 5.00 and 15.00 m.
Compare and explain. Do maximum distance and maximum flight time require the same conditions?
Model limit. Unequal heights or drag break the complementary-angle result.
04 · Is 45° always optimal?
An elevated launch has extra falling time, so its maximum-range angle is below 45°.
\tan\theta_\mathrm{opt}=\frac{v_0}{\sqrt{v_0^2+2gh}}
Try itFor h=20 m and v=20 m/s, compare recorded trials at 35.3° and 45°. Then set h=0.
Prediction. The optimum is about 35.26°, with range 56.57 m; 45° gives about 54.64 m.
Compare and explain. How does the optimum angle change as launch height increases?
Model limit. The input step is 0.1°; compare attainable settings near the theoretical optimum.
05 · Changing gravity
With the same initial velocity, weaker gravity reduces vertical velocity more slowly.
R\propto\frac{1}{g},\qquad t_\mathrm{f}\propto\frac{1}{g}\quad(h=0)
Try itRecord 20 m/s at 45° with g=10, then g=5. Compare both paths on the shared scale.
Prediction. Halving g doubles range from 40 to 80 m and flight time from 2.83 to 5.66 s.
Compare and explain. Does flight time still exactly double for a nonzero launch height?
Model limit. Real worlds also differ in atmosphere and curvature; this controlled experiment changes only g.
06 · Energy exchange
During ascent kinetic energy falls and potential energy rises; their sum stays constant without drag.
K=\frac{1}{2}mv^2,\qquad U=mgy,\qquad E=K+U
Try itLoad 1 kg at 20 m/s and 30°. Use the energy graph and record the apex and just-before-impact states.
Prediction. Total energy is 200 J; at the apex K=150 J and U=50 J.
Compare and explain. Why are speeds equal at the same height during ascent and descent?
Model limit. Bounce and impact losses are omitted; the final velocity is the pre-impact value.
07 · Drag and asymmetry
Drag opposes instantaneous velocity and scales with speed squared; horizontal velocity is no longer constant.
\vec F_\mathrm{d}=-\frac{1}{2}\rho C_\mathrm{d}A\,v\vec v
Try itRecord a soccer ball at 30 m/s and 45° in vacuum. Enable drag and record the same launch again.
Prediction. Vacuum range is 90 m. With drag, range and peak height decrease and mechanical energy falls.
Compare and explain. Does drag still point downward during descent? Compare it with velocity.
Model limit. A constant drag coefficient omits spin, surface effects and speed-dependent aerodynamics.
08 · Separate mass and cross-section
For equal size, a larger mass suffers less drag acceleration. In vacuum mass does not change the path.
a_\mathrm{d}=\frac{\rho C_\mathrm{d}\pi r^2v^2}{2m}
Try itWith drag on, record 0.5 kg, then change only mass to 1 kg. Next hold mass fixed and change radius.
Prediction. Initial drag acceleration halves when mass doubles and quadruples when radius doubles. Whole-flight range does not scale by those same factors.
Compare and explain. Which variables change together in the ball presets, making causal comparisons harder?
Model limit. Independent mass and radius changes are controlled experiments, not necessarily real balls.