Average Speed is defined as the mean speed of any object observed over a period of time. The average Speed Formula is the formula used to find an object's average speed throughout the journey which is completed over a duration of time. Suppose a car travels from point A to b and in t hours and comes back in T hours then the average speed of both the journey is the total distance covered by the car divided by the total time taken by it. Thus, the average speed formula is the total distance dived by the total time taken by an object.
The average speed formula is used when we have various paths covered by an object and we have to find the average speed of the object in complete motion. Average Speed Formula is important for Class 9 students. In this article, we will learn about, average speed, its formula, examples, and others in detail.
Average Speed Definition
Average speed is the mean rate at which anything moves. It is a constant and predictable rate with which an object has completed its journey in ideal terms. Average Speed is a scalar quantity. It has no direction and is indicated by the magnitude. The average speed of any object is defined as,
Average Speed is the total distance covered divided by the total time taken to cover that distance.
Average speed is also defined as mean speed and is very useful for studying the motion of various objects.
What is Average Speed Formula?
The formula for Average speed of a body is equal to the total distance covered, divided by the total time taken. The average speed formula is expressed as follows,
Average Speed = Total Distance Covered / Total Time Taken
Average Speed Formula
Suppose an object moves at a speed of s and S and covers a distance of d and D units in the time interval t and T unit respectively. Then the average speed formula for the same is,
Average Speed = (d + D)/(t + T)
As average speed is also speed, hence it is a scalar quantity and is measured in km/hr, m/s, etc.
Derivation of Average Speed Formula
We can derive the average speed formula by studying the example discussed. Let's imagine a man journeys to his favorite hilly place at different speeds, v1, v2, v3, ...., and so on, over time intervals of t1, t2, t3 traveled,..., and so on. As a result, the total distance traveled is v1t1 + v2t2 + v3t3 + .... vntn. The total time taken is t1 + t2 + t3 + ..... + tn.
The average speed at which he completed his voyage is,
Vavg = (v1t1 + v2t2 + v3t3 + . . . . + vntn) / (t1 + t2 + t3 +. . .+ tn)
If assumed that the time spent in each period is the same, t1 = t2 = t3 = . . . . = tn, the following new equation is obtained:
Vavg = (v1 + v2 + v3 +. . . +vn) / n
The 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:
Given,
- Speed in first hour = 38 miles/hour
- Speed in second hour = 54 miles/hour
- Speed in third hour = 55 miles/hour
Total Distance Covered = (Speed in First Hour × 1 + Speed in Second Hour × 1 + Speed in Third Hour × 1)
∴ T D= (38 + 54 + 55) = 147 miles ⇢ (Equation 1)
Total Time = 1+ 1 + 1 = 3 Hours ⇢ (Equation 2)
Therefore,
Average Speed = Total Distance Covered / Total Time
∴ Average Speed = 147 / 3 ⇢ (from equation 1 and 2)
∴ Average Speed = 49 miles/hour
Thus, the average speed in the complete journey is 49 miles/hour.
Unit of Average Speed
Average Speed is nothing but the mean speed of an object in total motion. It is calculated in the same unit such as speed. Thus, the unit of average speed is, m/sec, km/hr, miles/hr, etc.
Dimensional Formula for the Average Speed is [LT-1].
Average Speed Formula with Two Speeds
If any object travels at two speeds say v1 and v2 then the average speed of the object is calculated using the formula,
Average Speed = 2v1v2 / (v1 + v2)
Derivation Average Speed Formula with Two Speeds
We can derive the average speed formula if two speeds are given by example, suppose an object goes from A to B, distance covered D m, time taken t, and comes back in time T, then
Speed from A to B (v1) = D/t...(1)
Speed from B to A (v2) = D/T...(2)
Average Speed (vavg) = (Total Distance)/(Total Time)
Total Distance = D + D = 2D
Total Time = t + T
vavg = 2D/(t+T)
Dividing by D in numerator and denominator we get,
vavg = 2/{(t/D) + (T/D)}
From eq. (1) and (2)
vavg = 2/{1/v1 + 1/v2}
vavg = 2v1v2/(v1 + v2)
Thus, the average speed formula is verified.
Now let's learn the same with the help of an example.
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 sec. Find his speed when he goes from point A to point B if his average speed for the entire journey is 49 meters per sec.
Solution:
Consider, 'a' be a speed from place A to place B.
Speed from place A to place B = 56 meters per sec
He covers the same distance in both directions.
Formula of Average speed,
Average Speed = (2mn) / (m + n)
where,
- m = speed from place A to place B
- n (speed from place A to place B) = 56 meters per hour
∴ 49 = (2 × m × 56) / (m + 56)
∴ 49(m + 56) = 112m
∴ 49m + 2744 = 112m
∴ 2744 = 112m - 49m
∴ 2744 = 63m
∴ m = 43.56 meters/sec
Thus, the speed of Rohit from A to B is 43.56 m/s
Average Speed Formula Special Cases
There are several cases where the direct formula is used to calculate the average speed and some of these important cases are discussed below:
Case 1
If an object travels with a speed of s1 for time t1, and the speed of s2 for time t2, the formula for average speed is,
Average Speed Formula = (s1t1 + s2t2)/(t1 + t2)
For "n" different speeds the Average speed formula using the above concept is,
Average Speed Formula = (s1t1 + s2t2 + ... + sntn)/(t1 + t2 + ... + tn)
Case 2
If an object covers different distances, d1, d2, d3, ... dn, in different time interval, t1, t2, t3, ..., tn resctively. Then the average speed is calculated using the formula,
Average Speed Formula = (d1 + d2 + d3 + ... + dn) / (t1 + t2 + t3 + .... + tn)
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Average Speed Example
Example 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?
Solution:
Total distance traveled in the first three hours is = Speed × Time
Distance = 50 × 3 = 150 miles
Total distance traveled in the next four hours is = Speed × Time
Distance = 60 × 4 = 240 miles
Total distance traveled in the last five hours is = Speed × Time
Distance = 80 × 5 = 400 miles
Then, total distance covered = 150 + 240 + 400 = 790 miles
Total Time taken = 3 + 4 + 5 = 12 hours
Average Speed = Total distance covered / Total Time taken
Average Speed = 790 / 12
Average Speed = 65.8333 miles/hours
Thus, the average speed in the journey is 65.8333 miles per hour.
Example 2: 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 A to place B = 40 miles per hours
Speed from B to place A = 40 + 25% 0f 40 = 50 miles per hours
Average Speed = (2mn) / (m +n)
Here, m = 40, n = 50
Average Speed = (2 × 40 × 50) / (40 + 50)
Average Speed = 44.44
Thus, the average speed of the entire journey is 44.44 miles per hour.