Equation of Motion

The velocity of an object can be calculated by integrating the acceleration, which is assumed to be constant, over time which follows directly from the definition of the acceleration: $$\vec{v}(t) = \int \vec{a}\,\mathrm{d}t = \vec{a}t + \vec{v}_0$$ In this equation, the integration constant has been identified as the initial velocity $\vec{v}_0$ of the object. In order to calculate its position in space, a second integration is required, leading to the following relation: $$\vec{r}(t) = \int \vec{v}(t)\,\mathrm{d}t = \frac{1}{2} \vec{a}t^2 + \vec{v}_0t + \vec{r}_0$$ Similar to the previous result, the integration constant turns now out to be the initial position \(\vec{r}_0\). This very important equation is called Equation of Motion. This equation describes the location of an object that is accelerated with a constant acceleration \(\vec{a}\). A very important result is the quadratic dependency of the position on the time which results in a parabolic shape of the space-time curvature. If the object moves on a straight line, the coordinate system can be always rotated in a way that the object moves along one axis. In this case, the 3D curvature can be replaced with a simple scalar equation which is easier to handle: $$s = \frac{1}{2}at^2 + v_0t+s_0$$ The physical quantities position, velocity, and acceleration can be visualized with the help of so-called \(a\)-\(t\), \(v\)-\(t\), and \(s\)-\(t\) diagrams. The following figure shows an overview of the three possible cases for a constant position, a constant velocity, and a constant acceleration. The equation of motion will be discussed in the following sections for objects moving in the gravitational field of the earth.