# Average Speed Formula

Understanding the rate at which travel takes place necessitates an understanding of average speed. The speed of travel may change from time to time. In this scenario, determining the average speed is critical in order to estimate the pace at which the voyage will be completed.

### Average Speed

The average speed formula is used to calculate the uniform rate, which is the rate at which anything moves at a constant and predictable rate. Average speed is a scalar quantity. It has no direction and is indicated by the magnitude. Let’s look at how to compute average speed, the formula for calculating average speed, and some solved cases of average speed.

It is the total distance covered divided by the total time taken to cover that distance.

**Average Speed Formula**

Average Speed = Total distance covered / Total Time taken

**Derivation**

Let’s imagine a man journeys to his favorite hilly place at different speeds, v

_{1}, v_{2}, v_{3}, …., and so on, over time intervals of t_{1}, t_{2}, t_{3 },…, and so on. As a result, the total distance travelled is: v_{1}t_{1}+ v_{2}t_{2}+ v_{3}t_{3 }+ …. v_{n}t_{n}. The total time taken is t_{1}+ t_{2}+ t_{3 }+ ….. + t_{n}.As a result, the average speed at which he completed his voyage is,

V

_{avg}= (v_{1}t_{1}+ v_{2}t_{2}+ v_{3}t_{3}+ . . . . + v_{n}t_{n}) / (t_{1}+ t_{2}+ t_{3}+. . .+ t_{n})If assumed that the time spent in each period is the same, t

_{1}= t_{2}= t_{3}= . . . . = t_{n}, the following new equation is obtained:V

_{avg}= (v_{1}+ v_{2}+ v_{3}+. . . +v_{n}) / nThe average speed is the arithmetic mean of the individual speeds, as can be seen.

**Example: An automobile journey takes 3 hours. In the first hour, it travels 38 miles, 54 miles in the second hour, and 55 miles in the third hour.**

**Solution:**

Speed in first hour = 38 miles/hour

Speed in second hour = 54 miles/hour

Speed in third hour = 55 miles/hour

During the three-hour ride, travel at three distinct speeds.

If one wishes to calculate the average speed for a three-hour journey, then, take the ratio of Total distance covered and total time taken. Therefore,

Total distance covered = (Speed in a first hour + Speed in second-hour + Speed in the third hour)

∴ Total distance covered = (38 + 54 + 55) = 142 miles ⇢ (Equation 1)

Total time is taken = 3 hours ⇢ (Equation 2)

Therefore, the formula of average speed,

Average Speed = Total distance covered / Total Time taken

∴ Average Speed = 147 / 3 ⇢ (from equation 1 and 2)

∴ Average Speed = 49 miles/hour

**Unit of Average speed **

If the average speed is calculated, unit in this case, you’ll see that it’s as follows:

= (2(m/s)(m/s)) / ((m/s) + (m/s))

By removing the common words, return to the unit of m/s.

In a different way,

If someone travels from point A to point B at a certain speed, say m miles per hour. The person travels from point B to point A at a different rate, say n meter per hour. Hence, the person crosses the same distance in both directions, but at different speeds. Then how to find average speed?

Use the formula, **Average Speed = (2mn) / (m + n)**

**Derivation**

Let’s say a particle travels variable distances s

_{1}, s_{2}, s_{3}, s_{4 }…., s_{n}in the same direction with varying velocities v_{1}, v_{2}, v_{3 }…., v_{n}. The time is equal to the division of distance by the time, so we get the following formula for this scenario,Total time taken ‘t’ = (s

_{1}/ v_{1}) + (s_{2}/ v_{2}) + (s_{3}/ v_{3}) + … + (s_{n}/ v_{n})Here, Speed = Distance / Time, so the formula of average speed is,

V

_{avg}= (s_{1}+ s_{2}+ ….) / ((s_{1}/ v_{1}) + (s_{2}/ v_{2}) +….)If assume that the variable distances are equal, s

_{1}= s_{2}, the formula becomes:= 2s / ((s / v

_{1}) + (s / v_{2}) + …)= 2v

_{1}v_{2}/ ((1 / v_{1}) + (1 / v_{2}))∴ V

_{avg}= 2v_{1}v_{2}/ (v_{1}+ v_{2})The idea that average speed is the harmonic mean of particular speeds is vividly expressed in this statement.

**Example: Rohit travels at a set speed from point A to point B. When he returns from place B to place A, he travels at a rate of 56 meters per hour. Find his speed when he goes from point A to point B if his average speed for the entire journey is 99 meters per hour.**

**Solution: **

Consider, ‘a’ be a speed from place A to place B.

speed from place A to place B = 56 meters per hour

He covers the same distance in both directions.

The formula of Average speed,

Average speed = (2mn) / (m + n)

Here, m= speed from place A to place B, n=speed from place A to place B = 56 meters per hour

∴ 99 = (2 × m × 56) / (m + 56)

∴ 99(m + 56) = 112m

∴ 99m + 5544 = 112m

∴ 5544 = 112m – 99m

∴ 5544 = 13m

∴ m = 426.46 meters/hour

**Distance**

Distance is defined as the total length of a path connecting two places.

Formula for the Distance is,

**Distance = Rate × Time**

**Real life Example of Average Speed**

- Swapnil is a buddy who just completed three books in the span of nine months. How did he pull it off?

By using a straightforward technique. Every day, he wrote 1,000 words. (This should take roughly 2 to 3 pages) He did it every day for 253 days in a row. Consider this method in comparison to the traditional idea of a writer hiding out in a cabin for weeks, writing like a madman to finish their novel. The cabin maniac has a high “maximum speed” – perhaps 20 or 30 pages each day. However, after a few weeks at such an unsustainable rate, either the book or the author will be finished. Swapnil’s top speed, on the other hand, was never close to that of the cabin’s insane writer. His average speed, though, increased significantly over the course of a year or two.

- Anyone can get a surge of inspiration, go to the gym, and work hard for a single workout. That’s the top speed. We spend a lot of time worrying about it. What was the intensity of your workout? What motivates you? How hard are you pushing yourself?

But what if averaged all of the days from the previous month? How many of those days involved some sort of physical activity? What happened in the last three months? Or the previous year? What has been your average speed?

If looked at it this way, one might notice that they were sick for a week, that they skipped the gym a couple of times after a long day at work, and that they were also on the road for two weeks. Suddenly, one will realize that while the top speed may be tremendous on occasion, the average speed is significantly lower than believed.

**Summary**

Average Speed is defined as the total distance covered divided by the total time taken to cover that distance.

Formula of Average Speed:

Average Speed = Total distance covered / Total Time taken

If someone travels from point A to point B at a certain speed, say m miles per hour. He travels from point B to point A at a different rate, say n meter per hour. He crosses the same distance in both directions, but at different speeds.

Formula of Average Speed:

Average Speed = (2mn) / (m + n)

### Sample Problems

**Question 1: Dhanraj drove 50 miles per hour for 3 hours, 60 miles per hour for 4 hours, and 80 miles per hour for 5 hours. How fast did he travel on average throughout the journey?**

**Answer:**

The total distance traveled in the first three hours is = Rate × Time

Distance = 50 × 3

∴ Distance = 150 miles

The total distance traveled in the next four hours is = Rate × Time

Distance = 60 × 4

∴ Distance = 240 miles

The total distance traveled in the last five hours is = Rate × Time

Distance = 80 × 5

∴ Distance = 400 miles

Then, Total distance covered = 150 + 240 + 400

∴ Total distance covered = 790 miles

Then, Total Time taken = 3 + 4 + 5

∴ Total Time taken = 12 hours

Here,

Average Speed = Total distance covered / Total Time taken

∴ Average Speed = 790 / 12

∴ Average Speed = 65.8333

As a result, the overall average speed is 65.8333 miles per hour.

**Question 2: Define Average speed and its formula.**

**Answer:**

Average speed is defined as the total distance covered divided by the total time taken to cover that distance.

Formula of Average Speed:

Average Speed = Total distance covered / Total Time taken

**Question 3: At a speed of 40 miles per hour, it takes 5 hours to travel from point A to point B. He returns from place B to place A with a 25% boost in speed. Calculate the average speed of the entire journey.**

**Answer:**

Speed from place A to place B = 40 miles per hour.

Speed from place B to place A = 50 miles per miles. (25% increased)

Average Speed = (2mn) / (m +n)

Here, m = 40, n = 50

Average Speed = (2 × 40 × 50) / (40 + 50)

∴ Average Speed = 44.44

As a result, the overall average speed is 44.44 miles per hour.

**Question 4: Relationship between Average speed, Total distance covered, Total Time taken.**

**Answer:**

Here,

Average Speed = Total distance covered / Total Time taken

Average speed directly proportional to Total distance covered and inversely proportional to Total time taken.

**Question 5: Is average speed a vector or a scalar quantity?**

**Answer:**

Average speed is a scalar quantity.

**Question 6: What exactly does the Average Speed Formula indicate?**

**Answer** :

The average speed formula, often known as v

_{avg}, is used to calculate the average speed of a body that varies continuously over time intervals. If we have the final velocity, v, and the initial velocity, u, we can calculate v_{avg}using the formula.

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