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:

\[ \mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt} \]

We have seen, by analyzing the velocity, that:

\[ \lim_{\Delta t \to 0} \frac{\Delta \mathbf{r}}{\Delta t} = \frac{d\mathbf{r}}{dt} = \mathbf{v} \]

Thus, we have:

\[ \mathbf{a} = \frac{d}{dt}\left( \frac{d\mathbf{r}}{dt} \right) = \frac{d^2 \mathbf{r}}{dt^2} \]

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:

\[ a_t = \frac{dv}{dt} = \mathbf{i} \, a_t \]

where \( v \) represents the magnitude of the velocity vector \( \mathbf{v} \) and \(\mathbf{i}\) represents a directed and oriented vector.

The acceleration vector consists of two parts: a tangential component and a normal component.
  • 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:

\[ y = \int_0^t v(t) \, dt = \int_0^t (v_0 + a_t t) \, dt \]

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.