6.3.4 Getting back to Displacement

Mathematically, we can derive displacement as $D=\int v(t)dt$, in which $v(t)$ is the velocity at time $t$. If the velocity does not change (constant velocity), then

$$D=vt$$ (6.1)

in which $v$ is the constant velocity, and $t$ is the amount of time that the robot travels at $v$. This equation is useful when the robot is traveling at a constant velocity.

Another way to write velocity is $v=\int a(t)dt$, in which $a(t)$ is the acceleration at time $t$. If we assume accleration is constant, then $v(t)=at$, in which $a$ is the constant acceleration, and $t$ is the amount time for acceleration.

Substituting $v(t)=at$ into $D=\int v(t)dt$, we can say that

$$D(t) = \int_0^t v(x)dx$$ (6.2)

$$= \int_0^t axdx$$ (6.3)

$$= a \int_0^t xdx$$ (6.4)

$$= \frac{at^2}{2}$$ (6.5)

$$= \frac{v(t)^2}{2a}$$ (6.6)

This equation is useful during acceleration.