Projectile Motion Simulator

Horizontal and vertical motion are independent. Gravity pulls downwards and does nothing at all sideways, so in the absence of air resistance the horizontal velocity never changes while the vertical velocity falls steadily at g metres per second every second. A projectile is therefore two simple problems happening at once, not one complicated one.

That single idea generates every formula in the topic. With launch speed v at angle theta, and no air:

• Horizontal: distance equals v cos theta multiplied by time. Constant speed, nothing else to say.
• Vertical: height equals launch height plus v sin theta multiplied by time, minus one half g time squared. Standard constant-acceleration motion.
• Flight time on level ground: 2 v sin theta divided by g, which is simply the time to slow to a stop vertically, doubled.
• Maximum height: v sin theta all squared, divided by 2g.
• Range on level ground: v squared times sin of 2 theta, divided by g.

Two consequences catch people out. First, a ball thrown horizontally and a ball dropped from the same height hit the ground at exactly the same moment, because their vertical stories are identical. Second, the range formula depends on sin of twice the angle, and sine is symmetric about 90 degrees, so 30 degrees and 60 degrees give the same range. Complementary angles always do.

Without air, a trajectory is a parabola and it is perfectly symmetric: the way up is a mirror image of the way down, the apex sits exactly halfway along the range, and the projectile lands at the speed it left. Switch drag on and every one of those statements fails.

• The path becomes lopsided. The descending half is steeper and shorter than the climbing half, because the projectile has lost horizontal speed it can never get back. The apex shifts past the halfway point.
• Impact speed drops below launch speed. Energy has gone into heating the air, and nothing returns it.
• Complementary angles stop matching. With drag, 30 degrees beats 60 degrees, because the steeper shot spends longer in the air being slowed down.
• Light objects suffer far more than heavy ones. Drag depends on frontal area while inertia depends on mass, so what matters is the ratio between them. That is why a table tennis ball and a shot put, launched identically, end up in completely different places.

Because drag depends on speed and speed depends on drag, there is no formula to solve. The simulator integrates the motion in four thousand steps using a fourth-order Runge-Kutta method and then refines the landing point by bisection. With drag switched off that same integrator reproduces the exact textbook answers to the last digit shown, which is a useful check to run with a class before trusting anything it says with drag on.

Two simplifications are worth stating plainly. The projectile is treated as a point with no spin, so there is no Magnus effect and no swinging cricket ball or curving free kick. Air density is held constant, which is fine near the ground but not for something fired several kilometres up.