Average Speed Formula

Average speed is the mean value of the speed of a body over a period of time. The formula for average speed is needed since the speed of a moving body is not constant and varies across a period of time. Even with varying speed, the values of total time and the total distance covered can be used, and with the help of the formula for average speed, we can find a single value to represent the entire motion.

What is the Average Speed Formula?

The average speed of a body is equal to the total distance covered, divided by the total time taken. The formula for average speed is given as: 

Average Speed Formula:

Average Speed = Total distance covered ÷ Total time taken

Average Speed Formula Special Cases 

Case 1: For a body or object traveling with a speed of \(s_1 \) for time \( t_1 \), and the speed of \(s_2 \) for time \( t_2 \), the formula for average speed is given in the below expression. The product of \(s_1 \times  t_1 \), and \(s_2 \times  t_2 \) gives the distances covered in time intervals \(t_1 \), and \(t_2 \) respectively.

Average Speed Formula \(= \frac{s_1 \times t_1 + s_2 \times t_2}{t_1 + t_2}\)

Case 2: Similarly, when 'n' different speeds, \(s_{1}, s_{2}, s_{3},... s_{n}\), are given for 'n' respective individual time intervals, \(t_{1}, t_{2}, t_{3},... t_{n}\) respectively, the average speed formula is given as:

Average Speed Formula \(= \frac{s_1 t_1 + s_2 t_2 + ... + s_n t_n}{t_1 + t_2 +...+ t_n}\) 

Case 3: Average speed when different distances,  \(d_{1}, d_{2}, d_{3},... d_{n}\), are covered for different intervals of time, \(t_{1}, t_{2}, t_{3},... t_{n}\) repectively is given as:

Average Speed Formula \(S_{avg}\) \(= \frac{d_1 + d_2 + d_3 +...+ d_n} {t_1 + t_2 + t_3 +....+ t_n}\)

Case 4: Average speed when different speeds, \(s_{1}, s_{2}, s_{3},... s_{n}\), are given for different distances, \(d_{1}, d_{2}, d_{3},... d_{n}\) respectively is given as:

Average Speed Formula \(S_{avg}\) \(= \frac{d_1 + d_2 + d_3 +...+ d_n} {\dfrac{d_1}{s_1} + \dfrac{d_2}{s_2} + \dfrac{d_3}{s_3} +....+ \dfrac{d_n}{s_n}}\)

Case 5: Average speed formula when two or more speeds are given (\(s_{1}, s_{2}, s_{3},... s_{n}\)) such that those speeds were traveled for same amount of time (\(t_{1} = t_{2} = t_{3} =... t_{n} = t)\) is given as:

Average Speed Formula, \(S_{avg}\) \(= \frac{s_{1} t + s_{2} t +...+ s_{n} t} {t\times n} = \frac{s_{1} + s_{2} +...+ s_{n}} {n} \)

Case 6: Average speed when different speeds given (\(s_{1}, s_{2}, s_{3},... s_{n})\) for same distance (\(d_{1} = d_{2} = d_{3} =... d_{n} = d)\) is given as:

Average Speed Formula \(S_{avg}\) \(= \frac{ n \times d} { d \times \left[ \dfrac{1}{s_1} + \dfrac{1}{s_2} + \dfrac{1}{s_3} +....+ \dfrac{1}{s_n}\right]} = \frac{ n} {\left[ \dfrac{1}{s_1} + \dfrac{1}{s_2} + \dfrac{1}{s_3} +....+ \dfrac{1}{s_n}\right]}\)

Examples on Average Speed Formula

Let us take a look at a few examples to better understand the formula for average speed.

Example 1: Using the average speed formula, find the average speed of Sam, who covers the first 200 kilometers in 4 hours and the next 160 kilometers in another 4 hours. 

Solution: 

To find the average speed we need the total distance and the total time.
Total distance covered by Sam = 200Km + 160 km = 360 km
Total time taken by Sam = 4 hour + 4 hour = 8 hour
Average Speed = Total distance covered ÷ Total time taken
Average Speed = 360 ÷ 8 = 45km/hr

Answer: Average speed of Sam is 45 km/hr.

Example 2: A train is moving with a speed of 80 miles per hour for the first 4 hours and 110 miles per hour for the next 3 hours. Find the average speed of the train with the help of the average speed formula.

Solution:

It is given that the train is moving at a speed of 80 miles per hour for the first 4 hours.
Here \(S_1 \) = 80 and \(T_1 \) = 4.
And the train is moving at a speed of 110 miles per hour for the next 3 hours.
Hence \(S_2 \) = 110 and \(T_2\) = 3.
Average Speed Formula = \(\frac{S_1 \times T_1 + S_2 \times T_2}{T_1 + T_2}\)
Average Speed = (80 × 4 + 110 × 3) ÷ (4 + 3)
= (650) ÷ (7) = 92.86 miles/hour

Answer: The average speed of the train is 92.86 miles/hour.

Example 3: A car travels at a speed of  45 km/hr for 5 hours and then decides to slow down to 40 km/hr for the next 2 hours. Calculate the average speed using the average speed formula.

Solution: 

Distance I = 45 × 5 = 225 miles
Distance II = 40 × 2 = 80 miles
Total distance = Distance 1 + Distance 2
D = 225 + 80 = 305 miles
Using average speed formula = Total distance traveled ÷ Total time taken
Average Speed = 305 ÷ 7 = 43.57 m/s.

Answer: Average speed of a car is 43.57 m/s.