AP Calculus AB and BC glossary

Initial velocity

Also called: Starting velocity

Initial velocity is an object's velocity at time zero. It is the initial condition that pins down the constant of integration when you antidifferentiate acceleration, so velocity at any later time equals the initial velocity plus the accumulated change in velocity since time zero.

Antidifferentiating acceleration pins down velocity only up to a constant, and the initial velocity is what determines that constant. If $A(t)$ is any antiderivative of $a(t)$ , then $v(t) = A(t) + C$ with $C = v(0) - A(0)$ , so the constant equals the initial velocity only when the antiderivative you wrote down is already zero at $t = 0$ , as $-32t$ is. Writing the accumulation form instead of a general antiderivative builds the condition in from the start, so there is no $C$ left to solve for.

v(t) = v(0) + \int_0^t a(u)\,du

Constant acceleration shows the role plainly. If $a(t) = -32$ , then $v(t) = -32t + v(0)$ , and the initial velocity is the vertical intercept of the velocity graph. An object thrown upward has $v(0) > 0$ and one released from rest has $v(0) = 0$ , which changes every later answer about direction, speed and total distance. Note also that a motion problem may hand you the condition at some time other than zero, and it does the same job.

The mistake

Antidifferentiating twice but keeping only one constant. Going from acceleration to position produces two of them, and the velocity constant survives into position as the linear term $v(0)t$ . Dropping it silently assumes the object started at rest, which is why a thrown ball and a dropped ball come out with identical position functions.

Appears in: Unit 4: Contextual Applications, Unit 8: Applications of Integration