Average Speed

When an object is moving with a variable speed, then the average speed of the object is that constant speed with which the object covers the same distance in a given time as it does while moving with variable speed during the given time. Average speed for the given motion is defined as the ratio of the total distance travelled by the object to the total time taken i.e.

Average Speed $ \displaystyle \text{= }\frac{{\text{total distance travelled}}}{{\text{total time taken}}}$

(i) If a particle travels distances $  \displaystyle {{S}_{1}},{{S}_{2}},{{S}_{3}}$ etc. with speeds $ \displaystyle {{v}_{1}},{{v}_{2}},{{v}_{3}}$ etc. respectively, in same direction then total distance travelled

Total distance travelled $ \displaystyle {={{S}_{1}}+{{S}_{2}}+{{S}_{3}}+…}$

Total time taken $ \displaystyle {=\frac{{{{S}_{1}}}}{{{{v}_{1}}}}+\frac{{{{S}_{2}}}}{{{{v}_{2}}}}+\frac{{{{S}_{3}}}}{{{{v}_{3}}}}+…}$

Average speed, $ \displaystyle {{{v}_{{av}}}\text{ }=\dfrac{{{{S}_{1}}+{{S}_{2}}+{{S}_{3}}+…}}{{\dfrac{{{{S}_{1}}}}{{{{v}_{1}}}}+\dfrac{{{{S}_{2}}}}{{{{v}_{2}}}}+\dfrac{{{{S}_{3}}}}{{{{v}_{3}}}}+…}}}$

If $ \displaystyle {{S}_{1}}={{S}_{2}}=S$ i.e. the body covers equal distances with different speeds then,

$ \displaystyle {{v}_{{av}}}=\dfrac{{2S}}{{S\left( {\dfrac{1}{{{{v}_{1}}}}+\dfrac{1}{{{{v}_{2}}}}} \right)}}=\dfrac{{2{{v}_{1}}{{v}_{2}}}}{{\left( {{{v}_{1}}+{{v}_{2}}} \right)}}$

It means average speed is equal to harmonic mean of individual speeds.

(ii) If a particle travels with speeds $ \displaystyle {{v}_{1}},{{v}_{2}},{{v}_{3}},$ etc. during intervals $ \displaystyle {{t}_{1}},{{t}_{2}},{{t}_{3}},$ etc. respectively, then

total distance travelled $ \displaystyle {={{v}_{1}}{{t}_{1}}+{{v}_{2}}{{t}_{2}}+{{v}_{3}}{{t}_{3}}+…}$

total time taken $ \displaystyle {={{t}_{1}}+{{t}_{2}}+{{t}_{3}}…}$

So average speed,

$ \displaystyle {{v}_{{av}}}=\dfrac{{{{v}_{1}}{{t}_{1}}+{{v}_{2}}{{t}_{2}}+{{v}_{3}}{{t}_{3}}+…}}{{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}…}}$

If $ \displaystyle {{t}_{1}}={{t}_{2}}={{t}_{3}}=…={{t}_{n}}=t\text{ (say)}$ then

$ \displaystyle \begin{array}{l}{{v}_{{av}}}=\dfrac{{\left( {{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+…} \right)t}}{{nt}}\\=\dfrac{{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+…}}{n}\end{array}$

It means average speed is equal to Arithmetic mean of individual speed.

Instantaneous speed

When they speed of an object is variable, then object possesses different speeds at different instants. The speed of an object at a given instant of time is called its instantaneous speed.

Let at an instant t, and object while moving covers a distance $ \displaystyle \Delta s$ in a small interval of time $ \displaystyle \Delta t$ around time t, so that $ \displaystyle \Delta t\to 0$, then

$ \displaystyle \text{Instantaneous speed }=\underset{{\Delta t\to 0}}{\mathop{{\lim }}}\,\dfrac{{\Delta s}}{{\Delta t}}=\dfrac{{ds}}{{dt}}$

where, $ \displaystyle \frac{{ds}}{{dt}}$ is the first derivative of distance with respect to time. In case of a uniform motion of an object, the instantaneous speed is equal to its uniform speed.