Introduction
Position:
s(t)
Velocity:
v(t)
Acceleration:
a(t)
v(t) = \frac{ds}{dt} \\ \text{} \\ a(t) = \frac{dv}{dt} = \frac{d^2 s}{dt^2}
Acceleration is the derivative of velocity.
Velocity is the derivative of position.
Therefore,
Position is the antiderivative of velocity.
Velocity is the antiderivative of acceleration.
v(t) = \int a(t) \, dt + C_1 \\ \text{} \\ s(t) = \int v(t) \, dt + C_2
C_1
is the initial velocity.
C_2
is the initial position.
Examples
Example 1: An object in free fall is accelerating at
-9.8 \, \text[m/s²]
.
Find its position function if
v(0) = 0
and
s(0) = 20 \, \text{m}
.
\begin{align*}
a(t) &= -9.8 \\\\
v(t) &= \int (-9.8) \, dt + C_1 \\\\
&= -9.8t + C_1 \\\\
v(0) &= 0 \\\\
C_1 &= 0 \\\\
s(t) &= \int (-9.8t) \, dt + C_2 \\\\
&= -4.9t^2 + C_2 \\\\
s(0) &= 20, \, C_2 = 20 \\\\
s(t) &= -4.9t^2 + 20
\end{align*}
Example 2: A particle accelerates at the rate:
a(t) = 2t + 1
.
v(0) = 2
and
s(0) = 5
.
Find
s(t)
.
\begin{align*}
a(t) &= 2t + 1 \\\\
v(t) &= \int (2t + 1) \, dt + C_1 \\\\
&= t^2 + t + C_1 \\\\
v(0) &= 2 \\\\
C_1 &= 2 \\\\
s(t) &= \int (t^2 + t + 2) \, dt + C_2 \\\\
&= \frac{t^3}{3} + \frac{t^2}{2} + 2t + C_2 \\\\
s(0) &= 5 \\\\
C_2 &= 5 \\\\
s(t) &= \frac{t^3}{3} + \frac{t^2}{2} + 2t + 5
\end{align*}