Acceleration

• The acceleration of an object is defined as the variation of the velocity $\vec{v}$ with respect to time $t$.
• The general form is given as $$\vec{a} = \frac{\mathrm{d}\vec{v}}{\mathrm{d}t}$$

The acceleration of an object that underlies a constant acceleration can be defined as the change of its speed with time: $$a = \frac{\Delta v}{\Delta t}$$ A negative acceleration leads to a decrease in speed and is therefore called deceleration.

The change in speed can be calculated according to $$\Delta v = a \Delta t$$ Since the speed increases linear with time, the distance $\Delta s$ is given as the area under the $v$-$t$ graph which has a triangular shape: $$\Delta s = \frac{1}{2}\Delta v\Delta t$$ Inserting the previous formula leads to the following correlation between distance and time: $$\Delta s = \frac{1}{2}a(\Delta t)^2$$ This equation indicates a parabolic shape of the $s$-$t$ diagram for constant accelerations.

For non-constant accelerations with an arbitrary direction, it becomes a vector and is then defined as the derivative of the velocity with respect to the time: Inserting the relation $$\vec{v} = \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}$$ leads to the following formula: $$\boxed{\vec{a} = \frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2}}$$ This equation states that the acceleration is given by the second derivative of the position vector.