Introduction
Kinematics is the study of the motion of objects, describing their position, velocity, and acceleration over time. Before diving into a detailed explanation of these phenomena, it is essential to introduce some key concepts.
- A material point is an idealized object whose size is considered negligible relative to the distances involved in its motion.
- The trajectory is the path traced by a material point as it moves through space.
- A motion is said to be rectilinear if its trajectory lies along a straight line.
If a material point moves along a straight-line path under constant acceleration, meaning that the rate of change of velocity remains uniform over time, the motion is called uniformly accelerated rectilinear motion.
Acceleration
Let us consider a particle moving along a straight-line trajectory, where the position as a function of time is not described by a linear equation. Let $P_{1}$ and $P_{2}$ denote two positions of the material point along the $x$-axis at times $t_{1}$ and $t_{2}$, respectively. We denote by $\mathbf{v}{1}$ and $\mathbf{v}{2}$ the corresponding velocity vectors, with $\mathbf{v}{1} \neq \mathbf{v}{2}$. The vector acceleration is defined as the following limit:
\[\mathbf{a} = \underset{\Delta t \rightarrow 0}{lim} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d \mathbf{v}}{d t}\]We have seen, by analyzing the velocity, that:
\[\underset{\Delta t \rightarrow 0}{lim} \frac{\Delta \mathbf{r}}{\Delta t} = \frac{d \mathbf{r}}{d t} = \mathbf{v}\]Thus, we have:
\[\mathbf{a} = \frac{d}{d t} ( \frac{d \mathbf{r}}{d t} ) = \frac{d^{2} \mathbf{r}}{d t^{2}}\]Starting from the general expression of acceleration it is possible to introduce the concept of tangential acceleration As a point $P$ travels along a given path, the acceleration vector $\mathbf{a}$ can be broken down into two components:
- One tangential to the trajectory.
- One normal to the trajectory (also called centripetal acceleration that points toward the center of the curvature of the path).
The tangential acceleration, denoted by $\mathbf{a}_{t}$, corresponds to the variation of the speed over time. It is defined as:
\[a_{t} = \frac{d v}{d t} = \mathbf{i} a_{t}\]where $v$ represents the magnitude of the velocity vector $\mathbf{v}$ and $\mathbf{i}$ represents a directed and oriented vector.
- If the magnitude of the velocity changes, there is tangential acceleration $( a_{t} \neq 0 )$.
- If the magnitude of the velocity remains constant, the tangential acceleration is zero $( a_{t} = 0 )$.
Uniformly accelerated motion is a type of motion in which the tangential acceleration $a_{t}$ is constant at every point and equal to the average acceleration over any time interval. We have:
\[\frac{v - v_{0}}{t - t_{0}} = a_{t}\]Starting from this formula, solving for $v$ and assuming $t_{0} = 0$, we obtain:
\[v = v_{0} + a_{t} t\]In this way, derived the expression for velocity based on the definition of acceleration. Starting from the expression of velocity as a function of time we can derive the equation of motion by integrating with respect to time:
\[y = \int_{0}^{t} v ( t ) d t = \int_{0}^{t} ( v_{0} + a_{t} t ) d t\]Evaluating the integral, we obtain:
\[y = v_{0} t + \frac{1}{2} a_{t} t^{2}\]where $y$ represents the displacement of the material point along the trajectory as a function of time.