Understanding the Average Velocity Formula

Average velocity is a fundamental concept in kinematics that quantifies the rate at which an object changes its position with respect to time. It is essential in the analysis of motion, especially in problems involving displacement and direction. Understanding the average velocity formula is important for accurately solving problems in JEE Main Physics and related examinations.

Definition of Average Velocity

Average velocity is defined as the ratio of total displacement to the total time taken. Unlike average speed, it accounts for direction and is a vector quantity. In mathematical terms, average velocity indicates how quickly and in which direction an object’s position changes over a time interval.

Average Velocity Formula

The formula for average velocity is given by the displacement divided by the total time interval. Displacement refers to the straight-line distance from the initial to final position, considering direction.

The average velocity ($v_{avg}$) is expressed as:

$v_{avg} = \dfrac{\text{Total Displacement}}{\text{Total Time Taken}}$

In symbols, if $\Delta x$ is the displacement and $\Delta t$ is the time interval, then $v_{avg} = \dfrac{\Delta x}{\Delta t}$.

For a journey with initial position $x_1$ at time $t_1$ and final position $x_2$ at time $t_2$:

$v_{avg} = \dfrac{x_2 - x_1}{t_2 - t_1}$

This expression is commonly used in solving kinematics problems. More explanations are covered under Average Velocity Explained.

Units of Average Velocity

The SI unit of average velocity is metre per second (m/s). Depending on the context, it can also be expressed in centimetre per second (cm/s) or kilometre per hour (km/h). Both average speed and average velocity use the same units, though their physical significance differs.

Comparison: Average Velocity vs Average Speed

Average speed and average velocity are related but distinct quantities. Average speed is the ratio of total distance travelled to total time, while average velocity uses displacement. The following table highlights the key differences.

It is important to note that the magnitude of average velocity can be less than or equal to average speed, and becomes equal only when displacement equals distance, such as straight-line motion without change in direction. For more distinctions, refer to Average Speed Formula.

Average Velocity in Uniform and Non-uniform Motion

In uniform motion, where velocity does not change, the average and instantaneous velocity are equal. In non-uniform motion, where velocity varies, average velocity is computed by considering total displacement and total time, regardless of the path or changes in direction.

Average Velocity for Constant Acceleration

When an object moves with constant acceleration, the average velocity during a time interval can be represented as the arithmetic mean of initial and final velocity.

$v_{avg} = \dfrac{u + v}{2}$

Here, $u$ is the initial velocity, and $v$ is the final velocity.

This result is derived from linear motion equations, as explained in detail in Motion in One Dimension.

Average Velocity Formula: Calculus Approach

In calculus-based physics, average velocity between $t_1$ and $t_2$ is calculated as the change in position divided by the time interval.

$v_{avg} = \dfrac{s(t_2) - s(t_1)}{t_2 - t_1}$

If the position function $s(t)$ is known, the average velocity over an interval can be found using this formula. For topics related to calculus and kinematics, see Introduction to Kinematics.

Solved Example: Average Velocity in a Two-Part Journey

A student walks 100 metres east in 2 minutes and then 50 metres west in 1 minute. Calculate the average velocity.

Total displacement = $100\,\text{m}$ (east) $- 50\,\text{m}$ (west) $= 50\,\text{m}$ east

Total time = $2\,\text{min} + 1\,\text{min} = 3\,\text{min} = 180\,\text{s}$

Average velocity $= \dfrac{50\,\text{m}}{180\,\text{s}} = 0.278\,\text{m/s}$ (east)

This worked example shows that average velocity depends on net displacement and total time. More practice questions are available at Kinematics Important Questions.

Key Points to Remember about Average Velocity

• Average velocity is a vector quantity
• Direction is determined by net displacement
• It can be positive, negative, or zero
• Magnitude is always less than or equal to average speed
• SI unit is m/s (metre per second)

Common Mistakes with Average Velocity Calculations

• Using distance instead of displacement
• Ignoring direction in answers
• Mixing different units for distance and time
• Omitting stepwise workings in solutions

To avoid errors, always clarify displacement and maintain consistent units. Showing calculations stepwise ensures full credit in examinations. Related concepts are illustrated in Displacement and Velocity Time Graphs.