Speed and Velocity

In linear or translational motion a whole body moves from one point to another which is different from rotational or angular motion where a whole body rotates about a fixed axis.

When simply talking about speed, velocity and acceleration, we are talking about linear speed, linear velocity and linear acceleration respectively.

We usually don't use the term linear or translational for position, speed, velocity and acceleration but while comparing with or using with rotational or angular motion we use linear or translational to distinguish with angular motion.

In straight line motion, that is one dimensional motion such as along x-axis, a single number could determine the position of a particle but in two dimensional motion such as in xy-plane the position is determined by position vector drawn from origin of coordinate system to the location of the particle. Therefore, vectors are extensively used in two or three dimensional motion and very good understanding of vectors is required.

In two or three dimensional motion the quantities such as displacement, velocity and acceleration no longer lie along straight line.

Linear Speed

The linear speed of a body is the total distance traveled by the body per unit time which doesn't matter whether the path of motion is a curve or a straight line.

We do not take any account of direction of motion in speed. The speed is also the magnitude of instantaneous velocity at any particular point of the path of motion.

To find the distance traveled per unit time you divide the total distance traveled $s$ by the total time taken $t$ to cover that distance. The speed is always a positive quantity.

\[{\rm{speed}} = \left| {\frac{s}{t}} \right|\]

You may think the speed and velocity are the same things in everyday life but this is not the case in Physics as there are physical quantities with or without direction.

Linear Velocity

The velocity is the rate of change of displacement. The displacement is different from distance and always has a direction and therefore a velocity is always associated with a direction.

In speed a body can move in multiple directions (it means it can turn in various directions during the motion) but in velocity the particle should move without turning or changing the direction.

So, velocity is a vector quantity and speed is a scalar quantity. Note that speed is always positive. If the direction of the particle changes not the magnitude of velocity, we still say that the velocity changes as you should always think both magnitude and direction for velocity.

Figure 1 A car moves form point ${{p}_1}$ to point ${{p}_2}$ in the time interval $\Delta {t}$

In Figure 1 a car is moving in a straight line from point ${{p}_{1}}$ to point $p_2$ along x-axis. The distance between the points is $\Delta s={{s}_{2}}-{{s}_{1}}$. Let the time at point $p_1$ is $t_1$ and at point $p_2$ is $t_2$, so the total time during the motion between the points is $\Delta t={{t}_{2}}-{{t}_{1}}$. Now the magnitude of velocity is the total distance travelled in a particular direction divided by the total time taken during the displacement, which is,

\[v=\frac{{{s}_{2}}-{{s}_{1}}}{{{t}_{2}}-{{t}_{1}}}=\frac{\Delta s}{\Delta t} \tag{1} \label{1}\]

Eq. \eqref{1} gives the average velocity between the points ${{p}_{1}}$ and ${{p}_{2}}$. When we simply say the term "velocity" we generally refer to the instantaneous velocity. Instantaneous velocity is the velocity at any instant of time during the motion.

For example, if you say the velocity of a car is $13\text{m/s}$ at a particular point of time, that velocity is instantaneous velocity. To find instantaneous velocity, let $\Delta t$ approaches zero in Eq. \eqref{1}. When $\Delta t$ approaches zero, the point $p_2$ moves closer and closer to the point $p_1$. So the magnitude of the instantaneous velocity in the limit of $\Delta t$ approaches zero is,

\[v=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{\Delta s}{\Delta t}=\frac{ds}{dt} \tag{2} \label{2}\]

Note that $\Delta t$ approaches zero means the difference ${{t}_{2}}-{{t}_{1}}$ approaches zero which also makes the displacement, $\Delta s$ very small.

Figure 2 The displacement-time graph for constant velocity.

In Figure 2 the displacement-time graph is a straight line where $s$ is the displacement and $t$ is the time. If the velocity is constant the s-t graph is always a straight line with constant slope.

You should know that when the term "velocity" is used we mean "instantaneous velocity". When the velocity changes the s-t graph is always a curve as in Figure 3 and the tangent at each point on the curve represents the slope at that point. The slope gives the instantaneous velocity or simply velocity at any point on the curve.

Figure 3 If the velocity changes the displacement time graph will be some kind of a curve and the tangent at every point on the curve gives the slope which is the instantaneous velocity.
Mechanics
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