Speed and Velocity
Category : JEE Main & Advanced
(1) Speed : The rate of distance covered with time is called speed.
(i) It is a scalar quantity having symbol \[\upsilon \].
(ii) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{-1}}]\]
(iii) Unit : metre/second (S.I.), cm/second (C.G.S.)
(iv) Types of speed :
(a) Uniform speed : When a particle covers equal distances in equal intervals of time, (no matter how small the intervals are) then it is said to be moving with uniform speed. In given illustration motorcyclist travels equal distance (= 5m) in each second. So we can say that particle is moving with uniform speed of 5 m/s.
(b) Non-uniform (variable) speed : In non-uniform speed particle covers unequal distances in equal intervals of time. In the given illustration motorcyclist travels 5m in 1st second, 8m in 2nd second, 10m in 3rd second, 4m in 4th second etc.
Therefore its speed is different for every time interval of one second. This means particle is moving with variable speed.
(c) Average speed : The average speed of a particle for a given 'Interval of time' is defined as the ratio of total distance travelled to the time taken.
Average speed \[=\frac{\text{Total distance travelled}}{\text{Time taken}}\] ; \[{{v}_{av}}=\frac{\Delta s}{\Delta t}\]
\[{{v}_{av}}=\frac{\text{Total distance covered}}{\text{Total time elapsed}}\]
\[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\] = \[\frac{{{\upsilon }_{1}}{{t}_{1}}+{{\upsilon }_{2}}{{t}_{2}}+{{\upsilon }_{3}}{{t}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\]
\[{{\upsilon }_{av}}=\frac{\text{Total distance covered}}{\text{Total time elapsed}}\]\[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\]
\[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{\frac{{{d}_{1}}}{{{\upsilon }_{1}}}+\frac{{{d}_{2}}}{{{\upsilon }_{2}}}+\frac{{{d}_{3}}}{{{\upsilon }_{3}}}+......}\]
\[{{v}_{av}}=\frac{\int{vdt}}{\int{dt}}\]
(d) Instantaneous speed : It is the speed of a particle at a particular instant of time. When we say "speed", it usually means instantaneous speed.
The instantaneous speed is average speed for infinitesimally small time interval (i.e., \[\Delta t\to 0\]). Thus
Instantaneous speed \[v=\underset{\Delta t\to 0}{\mathop{\lim }}\,\,\,\frac{\Delta s}{\Delta t}\]\[=\frac{ds}{dt}\]
(2) Velocity : The rate of change of position i.e. rate of displacement with time is called velocity.
(i) It is a vector quantity having symbol \[\vec{v}\].
(ii) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{-1}}]\]
(iii) Unit : metre/second (S.I.), cm/second (C.G.S.)
(iv) Types of velocity :
(a) Uniform velocity : A particle is said to have uniform velocity, if magnitudes as well as direction of its velocity remains same and this is possible only when the particles moves in same straight line without reversing its direction.
(b) Non-uniform velocity : A particle is said to have non-uniform velocity, if either of magnitude or direction of velocity changes or both of them change.
(c) Average velocity : It is defined as the ratio of displacement to time taken by the body
\[\text{Average velocity}=\frac{\text{Displacement}}{\text{Time taken}}\]; \[{{\vec{v}}_{av}}=\frac{\Delta \vec{r}}{\Delta t}\]
(d) Instantaneous velocity : Instantaneous velocity is defined as rate of change of position vector of particles with time at a certain instant of time.
Instantaneous velocity \[\vec{v}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\,\,\frac{\Delta \,\vec{r}}{\Delta t}\]\[=\frac{d\vec{r}}{dt}\]
(v) Comparison between instantaneous speed and instantaneous velocity
(a) instantaneous velocity is always tangential to the path followed by the particle.
When a stone is thrown from point O then at point of projection the instantaneous velocity of stone is \[{{\vec{v}}_{1}}\], at point A the instantaneous velocity of stone is \[{{\vec{v}}_{2}}\], similarly at point B and C are \[{{\vec{v}}_{3}}\] and \[{{\vec{v}}_{4}}\] respectively.
Direction of these velocities can be found out by drawing a tangent on the trajectory at a given point.
(b) A particle may have constant instantaneous speed but variable instantaneous velocity.
Example : When a particle is performing uniform circular motion then for every instant of its circular motion its speed remains constant but velocity changes at every instant.
(c) The magnitude of instantaneous velocity is equal to the instantaneous speed.
(d) If a particle is moving with constant velocity then its average velocity and instantaneous velocity are always equal.
(e) If displacement is given as a function of time, then time derivative of displacement will give velocity.
Let displacement \[\vec{x}={{A}_{0}}-{{A}_{1}}t+{{A}_{2}}{{t}^{2}}\]
Instantaneous velocity \[\vec{v}=\frac{d\vec{x}}{dt}=\frac{d}{dt}\,({{A}_{0}}-{{A}_{1}}t+{{A}_{2}}{{t}^{2}})\] \[\vec{v}=-{{A}_{1}}+2{{A}_{2}}t\]
For the given value of t, we can find out the instantaneous velocity.
e.g for \[t=0\],Instantaneous velocity \[\vec{v}=-{{A}_{1}}\] and Instantaneous speed \[|\vec{v}|\,={{A}_{1}}\]
(vi) Comparison between average speed and average velocity
(a) Average speed is a scalar while average velocity is a vector both having same units (m/s) and dimensions \[[L{{T}^{-1}}]\].
(b) Average speed or velocity depends on time interval over which it is defined.
(c) For a given time interval average velocity is single valued while average speed can have many values depending on path followed.
(d) If after motion body comes back to its initial position then \[{{\vec{v}}_{av}}=0\] (as \[\Delta \vec{r}=0\]) but \[{{v}_{av}}>0\] and finite as \[(\Delta s>0)\].
(e) For a moving body average speed can never be negative or zero (unless \[t\to \infty )\] while average velocity can be i.e. \[{{v}_{av}}>0\] while \[{{\vec{v}}_{a\upsilon }}\]= or < 0.
(f) As we know for a given time interval
Distance \[\ge \] |displacement|
\[\therefore \] Average speed \[\ge \] |Average velocity|
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