Finding Instantaneous Velocity

Calculus 1 · Axiom Academy

Finding Instantaneous Velocity Throw a ball straight up and ask the hard question: how fast is it moving at one exact instant? Watch average velocities close in on the answer. A ball is thrown straight up; its height (ft) after t seconds is s(t) = −16t² + 64t . Velocity is change in position over change in time — but at a single instant there s no interval. Three moves get you to the exact velocity anyway. Set a moment t , hit Go, and watch the ball climb and fall along its curve. Read its height and its speed as it moves — notice the velocity is never the same twice, and it hits exactly 0 at the very top. You can t measure velocity at one instant directly — so measure the average velocity over [2, 2+h] and shrink h. Drag h smaller and watch the secant s slope, the average velocity, slide toward the instantaneous velocity at the peak. Read the velocity at any moment Do that shrinking at every time and you get the derivative: v(t) = s (t) = 64 − 32t . Drag the moment and read the instantaneous velocity straight off the slope of the tangent line — positive on the way up, zero at the peak, negative on the way down. One idea, three moves: watch it , shrink the interval , read the slope . The limit of average velocities is the derivative v(t)=s'(t) — the instantaneous rate of change. The same move powers a speedometer , a marginal cost in economics, the current in a circuit, and the acceleration of anything that moves.

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