Average velocity of a body can be given by the arithmetic mean of initial velocity and final velocity for a gi

Question

Average velocity of a body can be given by the arithmetic mean of initial velocity and final velocity for a given period of time when the body is:

Examoogle · Accepted Answer

The correct answer is Option A: moving with uniform acceleration. The average velocity of a body moving with a uniform acceleration over a given time interval is simply equal to the arithmetic mean of its initial velocity and final velocity.

Step-by-Step Solution & Concepts:
The average velocity of a body moving with a uniform acceleration over a given time interval is simply equal to the arithmetic mean of its initial velocity and final velocity.

Working Principle: According to the first equation of motion, $v = u + at$. The displacement $s$ under uniform acceleration $a$ can be found using the second equation of motion, $s = ut + \frac{1}{2}at^2$. By definition, average velocity is $v_{avg} = \frac{s}{t} = \frac{ut + \frac{1}{2}at^2}{t} = u + \frac{1}{2}at$. Substituting $at = v - u$, we get $v_{avg} = u + \frac{1}{2}(v - u) = \frac{u + v}{2}$.

Key Formulas: $v_{avg} = \frac{u + v}{2}$ — Average velocity under uniform acceleration; $v = u + at$ — First equation of motion; $s = ut + \frac{1}{2}at^2$ — Second equation of motion

Key Points: The arithmetic mean formula for average velocity holds strictly when acceleration is constant (uniform).; If acceleration is non-uniform, the position-time relationship is non-linear, and the simple mean formula $v_{avg} = \frac{u+v}{2}$ is invalid.; For uniform motion (zero acceleration), velocity is constant, so $u = v = v_{avg}$.

Advantages: Simplifies kinematic calculations for uniformly accelerated motion such as free fall.

Disadvantages: Cannot be applied directly when acceleration varies with time or position.

Applications: Motion under gravity (free fall); Vehicles accelerating uniformly from rest or braking at a constant rate

Additional Info: Option A is correct because uniform acceleration ensures a linear velocity-time graph, making the average velocity exactly equal to the arithmetic mean of initial and final velocities.

Answer: A is correct — Average velocity is given by the arithmetic mean of initial and final velocities only when the body is moving with uniform acceleration.

Exam Tip: Remember that on a velocity-time graph, uniform acceleration is represented by a straight inclined line, and the area under this line represents displacement.

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