Current Affairs JEE Main & Advanced

The smallest part of matter with zero dimension which can be described by its mass and position is defined as a particle or point mass. If the size of a body is negligible in comparison to its range of motion then that body is known as a particle. A body (Group of particles) can be treated as a particle,  depends upon types of motion. For example in a planetary motion around the sun the different planets can be presumed to be the particles.    In above consideration when we treat body as particle, all parts of the body undergo same displacement and have same velocity and acceleration.  

(1) Distance : It is the actual length of the path covered by a moving particle in a given interval of time. (i) If a particle starts from A and reach to C through point B as shown in the figure. Then distance travelled by particle \[=AB+BC=7\]m (ii) Distance is a scalar quantity. (iii) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{0}}]\] (iv) Unit : metre (S.I.)     (2) Displacement : Displacement is the change in position vector i.e., A vector joining initial to final position. (i) Displacement is a vector quantity (ii) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{0}}]\] (iii) Unit : metre (S.I.) (iv) In the above figure the displacement of the particle \[\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}\] \[\Rightarrow \] \[|AC|\]        \[=\sqrt{{{(AB)}^{2}}+{{(BC)}^{2}}+2(AB)\,(BC)\,\cos {{90}^{o}}\,}\]= 5 m (v) If \[{{\vec{S}}_{1}},\,{{\vec{S}}_{2}},\,{{\vec{S}}_{3}}\,........\,{{\vec{S}}_{n}}\] are the displacements of a body then the total (net) displacement is the vector sum of the individuals. \[\vec{S}={{\vec{S}}_{1}}+\,{{\vec{S}}_{2}}+\,{{\vec{S}}_{3}}+\,........\,+{{\vec{S}}_{n}}\] (3) Comparison between distance and displacement : (i) The magnitude of displacement is equal to minimum possible distance between two positions. So distance \[\ge \]|Displacement|.    (ii) For a moving particle distance can never be negative or zero while displacement can be. (zero displacement means that body after motion has came back to initial position) i.e., Distance > 0 but  Displacement  > = or < 0 (iii) For motion between two points, displacement is single valued while distance depends on actual path and so can have many values. (iv) For a moving particle distance can never decrease with time while displacement can. Decrease in displacement with time means body is moving towards the initial position. (v) In general, magnitude of displacement is not equal to distance. However, it can be so if the motion is along a straight line without change in direction. (vi) If \[{{\vec{r}}_{A}}\] and \[{{\vec{r}}_{B}}\] are the position vectors of particle initially and finally. Then displacement of the particle \[{{\vec{r}}_{AB}}={{\vec{r}}_{B}}-{{\vec{r}}_{A}}\] and s is the distance travelled if the particle has gone through the path APB.

(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 \[=\frac{\text{Total distance travelled}}{\text{Time taken}}\] ;   \[{{v}_{av}}=\frac{\Delta s}{\Delta t}\]

• Time average speed : When particle moves with different uniform speed \[{{\upsilon }_{1}}\], \[{{\upsilon }_{2}}\], \[{{\upsilon }_{3}}\] ... etc in different time intervals \[{{t}_{1}}\], \[{{t}_{2}}\], \[{{t}_{3}}\], ... etc respectively, its average speed over the total time of journey is given as

\[{{v}_{av}}=\frac{\text{Total distance covered}}{\text{Total time elapsed}}\] \[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\] = \[\frac{{{\upsilon }_{1}}{{t}_{1}}+{{\upsilon }_{2}}{{t}_{2}}+{{\upsilon }_{3}}{{t}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\]

• Distance averaged speed : When a particle describes different distances \[{{d}_{1}}\], \[{{d}_{2}}\], \[{{d}_{3}}\], ...... with different time intervals \[{{t}_{1}}\], \[{{t}_{2}}\], \[{{t}_{3}}\], ...... with speeds \[{{v}_{1}},{{v}_{2}},{{v}_{3}}......\] respectively then the speed of particle averaged over the total distance can be given as

\[{{\upsilon }_{av}}=\frac{\text{Total distance covered}}{\text{Total time elapsed}}\]\[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+......}\]         \[=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}+......}{\frac{{{d}_{1}}}{{{\upsilon }_{1}}}+\frac{{{d}_{2}}}{{{\upsilon }_{2}}}+\frac{{{d}_{3}}}{{{\upsilon }_{3}}}+......}\]

• If speed is continuously changing with time then      

\[{{v}_{av}}=\frac{\int{vdt}}{\int{dt}}\] (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 more...