Motion from Acceleration

Calculus 1 · Axiom Academy

A dropped ball only knows one thing — gravity, −32 ft/s². Integrate it up to speed, then to position, and watch the fall. Gravity gives every dropped object the same acceleration, −32 ft/s² . Integrate once for velocity v(t) = −32t , again for height s(t) = −16t² + s₀ — three things you can now do with that ladder. Drop it and watch the integral build Pick a release height, hit Drop, and the ball falls for real — height comes from integrating gravity twice, speed from integrating once. The position curve s(t) = −16t² + s₀ is drawn live as it falls. Read off the landing — no stopwatch needed You don't have to watch it fall to know how it lands. Set s(t) = 0 to get the fall time, plug into v(t) = −32t for the impact speed. Drag the height and read both off the gauge. Throw it up — what does extra speed buy? Now add a launch speed v₀ from the ground. The same −32 still pulls it back, but the constant of integration changes everything. Drag the throw and watch the hang time and the peak — the apex grows faster than the speed. One ladder, three moves: watch it , solve it , weigh it . Anywhere you know acceleration and want motion — a dropped wrench , a launched rocket , a tossed ball — integrate once for velocity, again for position, and let the constants carry the starting speed and height.

This is the written version of the interactive lesson above. See the full Calculus 1 course.