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 at constant velocity, meaning that the distances traveled are proportional to the time intervals taken, the motion is called uniform rectilinear motion.
Velocity
Let’s consider two points, $x_{1}$ and $x_{2}$, representing the position $P$ of a point at two successive moments in time, $t_{1}$ and $t_{2}$, respectively.
We can express the following relationship:
\[\frac{x_{2} - x_{1}}{t_{2} - t_{1}} = v\]This relationship shows that the distance traveled is proportional to the elapsed time, and the ratio remains constant, equal to the magnitude $v$. The instantaneous scalar speed is defined as the limit of the ratio as the time interval approaches zero:
\[v_{s} = \underset{\Delta t \rightarrow 0}{lim} \frac{\Delta x}{\Delta t} = \underset{\Delta t \rightarrow 0}{lim} \frac{x ( t + \Delta t ) - x ( t )}{\Delta t}\]where $v$ represents the instantaneous speed, $\Delta x$ is the displacement, and $\Delta t$ is the time interval. Therefore, the instantaneous scalar speed is given by the derivative of $x = x ( t )$ with respect to time.
Let us now imagine that the position $P$ of the point at time $t$ is identified by the displacement vector $\overset{\rightarrow}{r}$. The displacement from $P$ to $P^{‘}$ occurs over a time interval $\Delta t$ and is represented by the vector $\Delta \overset{\rightarrow}{r}$. We have:
\[\underset{\Delta t \rightarrow 0}{lim} \frac{\Delta \mathbf{r}}{\Delta t} = \frac{d \mathbf{r}}{d t} = \mathbf{v}\]
Thus, we can define the velocity vector as:
\[\mathbf{v} = \frac{d x}{d t} = \mathbf{i} \frac{d x ( t )}{d t}\]where $\mathbf{i}$ represents a directed and oriented vector. The velocity vector is tangent to the trajectory at each point, oriented according to the direction of motion, and has a magnitude equal to the scalar speed.
- In uniform rectilinear motion, the velocity vector remains constant.
- In uniform rectilinear motion, the trajectory’s position-time equation is a straight line, meaning the position varies linearly with time.
Velocity is measured in units of length multiplied by time raised to the power of $- 1$, and its standard unit is meters per second $( \text{ms}^{- 1} )$.
Example
Imagine a car traveling along a straight road at a constant speed of $v = 20 m / s$. Since the velocity is constant, the distance traveled by the car is directly proportional to the elapsed time. The position $x ( t )$ of the car at any time $t$ can be expressed as:
\[x ( t ) = x_{0} + v t\]where:
- $x_{0}$ is the initial position (at $t = 0$).
- $v$ is the constant velocity.
- $t$ is the time elapsed.
This means that for every second that passes, the car moves exactly 20 meters forward, without speeding up or slowing down. Let’s summarize the data in a table:
\[\text{Time} ( s ) & \text{Position} ( m ) \\ 0 & 0 \\ 1 & 20 \\ 2 & 40 \\ 3 & 60 \\ 4 & 80 \\ \vdots & \vdots\]Glossary
- Kinematics: the study of the motion of objects, describing their position, velocity, and acceleration over time.
- Material point: an idealized object whose size is considered negligible relative to the distances involved in its motion.
- Trajectory: the path traced by a material point as it moves through space.
- Rectilinear motion: motion whose trajectory lies along a straight line.
- Uniform rectilinear motion: motion along a straight line with constant velocity, where distances traveled are proportional to time intervals.
- Scalar speed: the limit of the ratio of displacement to time interval as the time interval approaches zero; the magnitude of the velocity vector.
- Velocity vector: the rate of change of the displacement vector with respect to time; a vector tangent to the trajectory, oriented in the direction of motion, with a magnitude equal to the scalar speed.
What is velocity
- Velocity describes how fast and in what direction an object moves.
- Scalar velocity refers to the absolute value of velocity, representing only the speed of the object without considering the direction.
- Vector velocity is the rate of change of position with respect to time, expressed as a vector tangent to the trajectory and oriented in the direction of motion.