Speed and Velocity

Speed

Speed of an object is defined as the time rate of change of position of the object in any direction. It is measured by the distance travelled by the object in unit time in any direction. i.e.,

$ \displaystyle \text{Speed = }\dfrac{{\text{distance travelled}}}{{\text{time taken}}}$

Speed is a scalar quantity. It gives no idea about the direction of motion of the object. The speed of the object can be zero or positive but never negative. The unit of speed is cm/s in CGS system and m/s in MKS system or S.I. The dimensional formula of speed is $ \displaystyle \left[ {{{M}^{0}}{{L}^{1}}{{T}^{{-1}}}} \right]$.

1. Uniform speed

And object is said to be moving with a uniform speed, if it covers equal distance in equal intervals of time, howsoever small these intervals maybe.

2. Variable speed

And object is said to be moving with a variable speed if it covers equal distance is unequal intervals of time or unequal distance in equal intervals of time, howsoever small these interval maybe.

3. 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.

4. 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.

Velocity

Velocity of an object is defined as a time rate of change of displacement of the object. It is also define as the speed of an object in a given direction. Quantitatively,

$ \displaystyle \text{velocity = }\dfrac{{\text{displacement}}}{{\text{time interval}}}$

Velocity is a vector quantity as it has both, the magnitude (speed) and direction. The velocity of an object can be positive, zero and negative according displacement is positive, zero or negative.

The unit of velocity is cm/s in CGS system and m/s in MKS system or S.I. The dimensional formula of velocity is $ \displaystyle \left[ {{{M}^{0}}{{L}^{1}}{{T}^{{-1}}}} \right]$.

1. Uniform velocity

And object is said to be moving with a uniform velocity if it undergoes equal displacement in equal intervals of time howsoever small this intervals maybe.

2. Variable velocity

And object is said to be moving with a variable velocity if it undergoes equal displacement in unequal intervals of time or unequal displacement in equal intervals of time or changes direction of motion while moving with a constant speed.

3. Average velocity

It is that uniform velocity with which the object will cover the same displacement in same interval of time as it does with its actual variable velocity during that time interval.

4. Instantaneous velocity

The average velocity of a particle during a time interval cannot tell us how fast, or in what direction, the particle was moving at any given time during the interval. To describe the motion in greater detail we need to define the velocity at any specific instant of time or specific point along the path. Such a velocity is called instantaneous velocity it needs to be define carefully.

We know that average velocity is the ratio of total displacement upon total time taken i.e.

$ \displaystyle {{\overrightarrow{v}}_{{av}}}=\dfrac{{\text{total displacement}}}{{\text{total time taken}}}$

Instantaneous velocity is the limit of the average velocity as the time interval approaches zero; it equals the instantaneous rate of change of position with time. We use the symbol $ \displaystyle v$ with no subscript for instantaneous velocity:

$ \displaystyle v=\underset{{\Delta t\to 0}}{\mathop{{\lim }}}\,\dfrac{{\Delta x}}{{\Delta t}}=\dfrac{{dx}}{{dt}}$

We always assume that the time interval $ \displaystyle {\Delta t}$ is positive so that $ \displaystyle v$ has the same algebraic sign as $ \displaystyle {\Delta x}$

If the positive x-axis points to the right, a positive value of $ \displaystyle v$ means that $ \displaystyle x$ is increasing and the motion is towards the right; negative value of $ \displaystyle v$ means that $ \displaystyle x$ is decreasing and the motion is towards the left. A body can have positive $ \displaystyle x$ and negative $ \displaystyle v$ are the reverse; $ \displaystyle x$ tells us where the body is, while $ \displaystyle v$ tell us how its moving.

Instantaneous velocity, like average velocity is a vector quantity.