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Letâ€™s suppose that you are travelling to a new city. The new city is 300 km far and you reach there in 3 hrs. In another scenario, you reach in 2 hrs.

Do you find the difference between the two scenarios? Yes, of Of course! In the first scenario, you took more time than the second one. So, the speed is something that tells you how early you can reach your destination in minimum time.

So, the speed equation is:

SpeedÂ =Â Distance/Time

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The car is traveling from point A to point B, so its speed formula will be:

Speed = Distance travelled from point A to B divided by the time taken.

Several types of speeds are as follows:

Uniform speed

Variable speed

Instantaneous speed

Average speed

Relative Speed

We know that speed is a scalar quantity and it gives no idea of what direction a vehicle may take; however, it tells us the magnitude.

Speed and its Units

The unit of speed is m/s. In CGS (Centi-Gram-Second) system, the unit of speed is cm/s. For finding the dimensional formula of speed we have the following steps:

The dimensional formula of time = [T]

Distance in meter = [L]

We know that the speed formula is distance per unit time, so the dimensional formula of speed is [LT^{-1}].

Uniform Speed

If any object covers equal distances in equal intervals of time, it means the body is said to be moving with a uniform speed, whatever small intervals maybe.

Variable or Non-Uniform Speed

If an object travels unequal distances or in unequal intervals of time. The object is said to possess variable speed. A car is said to have a non-uniform speed.

Average Speed

When an object moves with a non-uniform speed; however, the average speed is that constant speed with which the object covers the same distance in a given time as it does while moving with a variable/continuously changing speed during the given time.

So, we define the average speed as the ratio of the total distance covered by the vehicle to the total time taken. The formula is given by:

Average speed =Â Total distance travelled upon the total time taken

If a particle travels distances viz: s1, s2, s3, â€¦â€¦.with speeds viz: v1, v2, v3,.....The total time taken will be:

\[\frac{s_{1}}{v_{1}} + \frac{s_{2}}{v_{2}} + . . . . \]

If the total distance travelled is:

s1 +Â s2 + s3

The total time taken is:

\[\frac{s_{1}}{v_{1}} + \frac{s_{2}}{v_{2}} + \frac{s_{3}}{v_{3}}\]

Then, the average speed is:

\[V_{av} = \frac{s_{1} + s_{2} + s_{3} . . .}{\frac{s_{1}}{v_{1}} + \frac{s_{2}}{v_{2}} + . . . .}\]

If the vehicle travels equal distances with different speeds then, the average speed equation is:

\[V_{av} = \frac{2s}{\frac{2}{v_{1}} + \frac{s}{v_{2}} + . . . . } = \frac{2v_{1}v_{2}}{v_{2} + v_{1}}\]

Here, we can see that the average speed is the harmonic mean of individual speeds.

If a particle travels with different speeds viz: v1, v2, v3,....and in unequal intervals of time viz: t1, t2, t3,.....So, the total distance travelled is:

So, the average speed will be:

\[\frac{v_{1}t_{1} + v_{2}t_{2} + v_{3}t_{3} . . .}{t_{1} + t_{2} + t_{3} . . . }\]

Now, letâ€™s say the time interval is equal so,Â t1 =Â t2 =Â t3 = n, so the formula becomes:

\[V = \frac{v_{1} + v_{2} + v_{3}}{n}\]

In this case, the average speed is the arithmetic mean of all the speeds.

An object is covering an equal distance in unequal intervals of time. So, the different speeds at different instants can be calculated by using the concept of instantaneous speed. So, here is the formula for the same:

Let at an instant t, an object covers a distance Î”s in a small interval of time Î”t. So, the instantaneous speed becomes:

\[\text{Instantaneous speed } = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\]

In case we find the first derivative of the above equation, we rewrite the equation as:

\[\text{Instantaneous speed } = \lim_{\Delta t \rightarrow 0} \frac{ds}{dt}\]

One must note that in the uniform motion, the instantaneous speed becomes equal to the uniform speed.

Relative Speed

If the two objects travel towards each other, then their speeds are added, i.e.,

vREV Â = Â v1 + v2

If they travel in the same direction, then the difference in the two speeds will be the relative speed, i.e.,

vREV Â = Â v1Â -Â v2

We observe the movement of people, dogs, the motion of electric motors, the rotation of the turbine, and many more objects in our day-to-day lives, where each of these has some speed associated with which they cover a distance in a certain amount of time.

Speed is the rate at which an object covers some distance in a specific amount of time. However, how much distance travelled or at what speed a car or any vehicle could cover a specific distance is calculable. Now, letâ€™s take an example for the measurement of the speed of a moving object:

Example: A bus travels from point P to Q at a speed of 40 kmph and comes back at a speed of 30 kmph. What is its average speed?

Solution: Given v1 = 40 kmph, v2 = 30 kmph

The time taken from A to B is:

t1 = s/40

And,

t2 = s/30

The total tie taken is:

t1 + Â t2 = s/40 + s/30

Â Â Â Â Â Â Â Â Â Â Â Â =Â 7s/120 s

We know that the formula for an average velocity is:

\[V_{av} = \frac{s_{1} + s_{2}}{t_{1} + t_{2}} = 2s \times 120/7 = 240/7\] or 34.3 kmph

FAQ (Frequently Asked Questions)

Question 1: Dispur is at a Distance of 350 km from Kerala. A Train Reaches Kerala at a speed of 30 kmph and Another Train Starts at the Same Time from Dispur at a Speed of 40 kmph. When will These Two Trains Meet?

Answer: Here, the relative speeds of two trains is 30 - (- 40) = 70 kmph

Time is taken to cover a distance of 350 km is:

350/70 = 5hrs

Hence, they will meet after 5 hrs from the start.

Question 2: An Aircraft that Travels at the Speed of 600 kmph Ejects its Products of Combustion at the Speed of 1800 kmph Relative to the Aircraft. What is the Speed of Projection with Respect to the Observer?

Answer: Given data:

v_{REV} = 1800 kmph

v_{A}Â =Â 600 kmph

To find:

v_{p} = ?

Solution:

v_{REV} Â = v_{p} - Â v_{A}

=Â 600 - 1800

â‡’ v_{p} = - 1200 kmph

Here, the negative sign shows that the ejection occurs in the opposite direction.

Question 3: Can an Object have a Constant Velocity but Changing Speed?

Answer: No, a body cannot have a constant velocity but have a changing speed, i.e.,Â

Velocity = Speed + direction.

Question 4: What is the Difference Between Speed and Velocity?

Answer: Speed is a scalar quantity (cannot determine the direction), while velocity is a vector quantity.