What Is the Average Speed Formula?

Average speed is a fundamental concept in kinematics that quantifies how fast an object traverses a certain distance over a given time interval. It is essential for analyzing motion when speed is not constant, and its calculation is important for solving a variety of problems in physics and competitive examinations.

Definition and Mathematical Expression of Average Speed

Average speed is defined as the total distance travelled divided by the total time taken for the journey. The formula may be written as $ \text{Average Speed} = \dfrac{\text{Total Distance}}{\text{Total Time}} $. This calculation provides a scalar quantity and is independent of the direction of motion.

Unit Analysis of Average Speed

The SI unit of average speed is metres per second (m/s), although it can also be expressed in kilometres per hour (km/h) as per the problem's requirements. Conversion between these units is often necessary in physics calculations.

General Formula for Average Speed

The basic formula for average speed when an object covers a total distance $d$ in total time $t$ is given by $ S_{\text{avg}} = \dfrac{d}{t} $. This equation is applicable to all types of motion regardless of varying speed during different segments of the journey.

Formulas for Average Speed in Multiple Segments

For journeys divided into segments with different speeds and times, the following formulas apply. These equations account for different time intervals or distances covered at varying speeds.

• Two speeds $s_1$ for time $t_1$, $s_2$ for time $t_2$: $ S_{\text{avg}} = \dfrac{s_1 t_1 + s_2 t_2}{t_1 + t_2} $
• n speeds $s_i$ for times $t_i$: $ S_{\text{avg}} = \dfrac{\sum_{i=1}^{n} s_i t_i}{\sum_{i=1}^{n} t_i} $
• n distances $d_i$ in times $t_i$: $ S_{\text{avg}} = \dfrac{\sum_{i=1}^{n} d_i}{\sum_{i=1}^{n} t_i} $

When equal distances are covered at different speeds $s_1$, $s_2$, the average speed is calculated using the harmonic mean, which is $ S_{\text{avg}} = \dfrac{2 s_1 s_2}{s_1 + s_2} $ for two segments.

Solved Example: Average Speed with Varying Speeds

A vehicle travels at 60 km/h for 2 hours and then at 80 km/h for 3 hours. The total distance covered is $60 \times 2 + 80 \times 3 = 120 + 240 = 360$ km. The total time taken is $2 + 3 = 5$ hours. Applying the formula, $ S_{\text{avg}} = \dfrac{360}{5} = 72 $ km/h.

Special Case: Average Speed for Equal Distances

If an object travels equal distances at two different speeds, $s_1$ and $s_2$, the average speed is not simply the arithmetic mean. Instead, it is given by $ S_{\text{avg}} = \dfrac{2 s_1 s_2}{s_1 + s_2} $. This arises from the differing time intervals required at each speed.

Average Speed versus Average Velocity

Average speed depends on the entire path length travelled and is always a positive scalar. In contrast, average velocity depends on the net displacement and is a vector quantity. For journeys where the initial and final positions are the same, average velocity may be zero while average speed remains positive.

Practical Insights and Exam Recommendations

While calculating average speed, always use total distance and total time. Never simply average individual speeds unless the time intervals are identical. Convert units as required for consistency in calculations. For comprehensive practice, refer to resources such as the Kinematics Mock Test available on Vedantu.

Common Mistakes in Average Speed Calculations

A frequent mistake is to directly average speeds without considering the conditions. It is essential to distinguish between the arithmetic and harmonic mean, depending on whether the intervals are equal in distance or time. Attention must also be paid to unit consistency throughout the calculation process.

Applications of Average Speed in Physics

Average speed concepts are applied in solving multistage motion problems involving trains, cars, or projectiles. They are also critical for interpreting graphical data and analysing race or pursuit scenarios. Further details can be explored in Kinematics Important Questions.

Conversions between Speed Units

For consistent results, convert all speed measurements to the same unit system before applying average speed formulas. The conversion between km/h and m/s is performed by using $1\, \text{km/h} = \dfrac{5}{18}\, \text{m/s}$. Mastery of such conversions is crucial for precise answers in examinations.

Summary Table: Key Average Speed Formulas

Further Practice and Related Topics

Further proficiency in average speed problems can be developed through practice tests and exploring related concepts. Topics such as the Speed of Sound Waves in Air provide additional context for understanding speed in physical systems. Additional mock test resources include Kinematics Mock Test 1, Kinematics Mock Test 2, and Kinematics Mock Test 3.