2D Kinematics Simulation: Independence of Axes in Projectile Motion
Drop a ball from a table and fire another horizontally from the same height at the same instant, and they hit the ground simultaneously — always. Horizontal and vertical motion are independent. Vertically both balls start with v₀ᵧ = 0 and fall under gravity alone, so Δy = ½gt² gives a fall time that depends only on the drop height and g. The fired ball's horizontal velocity carries it sideways but does nothing to speed up or slow down its fall.
Projectile Motion: Why Horizontal Speed Doesn't Change Fall Time
A bowling ball and a cannonball leave the wall top at the same instant: one dropped, one fired horizontally at v0. Mass never enters — every ball falls with the same g. Both start with vy = 0 and both feel the same g, so their heights agree at every instant — the rungs between the ghost images stay level — and they land together after tfall = √(2h/g). v0 appears nowhere in that expression: throw at 1 m/s or 100 m/s and the fall takes the same time.
Horizontal and vertical motion are independent. After launch nothing pushes the ball sideways, so vx stays at v0 and the horizontal gaps between ghosts are equal; gravity acts only downward, so vy grows by 9.8 m/s every second, exactly as for the dropped ball, and the vertical gaps grow. The throw decides only how far sideways the ball gets in that fixed time: Δx = v0·tfall. To hit the dragon, choose v0 = d / tfall — tfall is in the box above the scene, and a landing within 1.8 m of the dragon's centre counts.
Model: no air resistance, level ground. With drag the thrown ball, moving faster through the air, meets a larger resistive force whose vertical part slows its fall slightly, so it lands a little after the dropped one — the small print behind Galileo's “always”.
On the vertical axis, once both balls leave the table gravity is the only force acting on them, assuming no air resistance. Both start with an initial vertical velocity of zero, so their fall time comes entirely from Δy = ½gt². Same height, same g, identical time — the horizontal launch never enters that equation.
On the horizontal axis, the fired ball keeps its initial velocity vₓ because no force acts on it sideways after launch. That constant speed sets the range, Δx = vₓ × t, which is how far it travels before landing. Range and fall time are set by different equations, which is exactly why changing one leaves the other untouched.
Launch the cannonball slowly or at blinding speed: the vertical vectors map perfectly onto each other and both objects strike the floor at the same moment. This is the result Galileo used to break the Aristotelian idea that heavier or faster-moving objects fall differently, and it is the foundation of every projectile problem you will solve — you always split the motion into two independent one-dimensional problems and share only the time between them.
0 comments