DISTANCE,DISPLACEMENT,SPEED & VELOCITY
Speed: The distance travelled by the body in unit time is called its speed.
Speed (V) = \(\frac{{Dis\tan ce\,\,travelled}}{{Time\,\,taken}}\)
* speed is a scalar quantity.
* it is represented by v or u
units: CGS Unit : cm/s, SI unit: m/s
UNIFORM MOTION
If an object moving along the straight line covers equal displacement in equal intervals of time, i.e object moving with uniform velocity is said to be in uniform motion along a straight line. The following Fig. shows the position time graph of such a motion.
UNIFORM SPEED
If a particle moving along a straight line (say x-axis) travels equal distances in equal intervals of time however small the intervals may be, then the particle is moving with uniform speed.
NON-UNIFORM SPEED OR VARIABLE SPEED
If a particle moving along a straight line travels unequal distances in equal intervals of time or equal distances in unequal intervals of time, then it is said to be moving with nonuniform speed.
Eg: i) Motion of a freely falling body.
Eg: ii) Motion of a body thrown vertically upwards
AVERAGE SPEED
For a particle in motion (uniform or nonuniform), the ratio of total distance travelled to the total time of motion is called average speed.
\({\text{Average speed }} = \frac{{{\text{Total distance travelled}}}}{{{\text{Total time}}}}\)
\(= \frac{{{x_2} - {x_1}}}{{{t_2} - {t_1}}}\)
Where x1 and x2 are the positions of an object at the instants t1 and t2 respectively.
If \({S_1},{S_2},{S_3}......{S_n}\) are the distances travelled by a particle in the time intervals \({t_1},{t_2},.....{t_n}\)...... respectively then,
Average Speed=\(\frac{{{S_1} + {S_2} + {S_3} + ...... + {S_n}}}{{{t_1} + {t_2} + ,.... + {t_n}}}\)
INSTANTANEOUS SPEED
The speed of a particle at a particular instant of time is called it's instantaneous speed. (or)
It is also defined as the limit of average speed as the time interval (\(\Delta t\)) becomes infinitesimally small.
If is the distance travelled by a particle in a time interval then
Speed=\(\frac{{\Delta x}}{{\Delta t}}\)
If the time interval is chosen to be very small, i.e., as \(\Delta t \to 0\), then the corresponding speed is called instantaneous speed.
\(\therefore \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta x}}{{\Delta t}} = \frac{{dx}}{{dt}}\) =instantaneous speed
Non - uniform speed: If a body travels unequal distances in equal intervals of time (or)equal distance in unequal intervals of time the body is said to be travelling with non uniform (or) variable speed.
Application-1:
A body travelling between two positions travels first half of the distance with speed v1, and the next half of the distance with speed v2
The average speed of total motion is \(\frac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}\)
Let x be the total distance between two positions.
Let t1 be the time for first half and t2 be the time for the next half of the distance
Application-2:
A body is travelling between two positions A, B. It travels from A to B with speed v1 and then from B to A in the same path with speed v2 . The average speed of total motion is \(\frac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}\)
Application-3:
A body is travelling between two positions. The total distance is divided into n equal parts. These parts are travelled with speeds \({v_1},{v_2},.....{v_n}\) respectively. The average speed of total motion is such that
\(\frac{n}{{{\text{Average speed }}}} = \frac{1}{{{v_1}}} + \frac{1}{{{v_2}}} + .... + \frac{1}{{{v_n}}}\)
Application-4:
A body travelling between two positions travels with speed v1 for time t1 and then with speed for time For the total motion,
\({\text{Average speed }} = \frac{{{v_1}{t_1} + {v_2}{t_2}}}{{{t_1} + {t_2}}}\)
Application-5:
A body travelling between two positions travels first half of the time with speed v1 and the next half of the time with speed v2. The average speed of total motion is=\(\frac{{{v_1} + {v_2}}}{2}\)
Application-6:
A body travelling between two positions travels for the time intervals \({t_1},{t_2},.....{t_n}\)with speeds \({v_1},{v_2},.....{v_n}\) respectively
\({\text{Average speed }} = \frac{{{\text{Total distance travelled}}}}{{{\text{Total time}}}}\)
=\(\frac{{{v_1}{t_1} + {v_2}{t_2} + ..... + {v_n}{t_n}}}{{{t_1} + {t_2} + ,.... + {t_n}}}\)