In the formula for uniformly accelerated motion, average velocity = (u + v) / 2, what does the symbol u represent?

Introduction / Context:
In one dimensional motion with constant acceleration, several useful kinematic equations relate displacement, velocity, acceleration and time. One of these is the formula for average velocity when acceleration is uniform: average velocity equals half the sum of initial and final velocities. In this formula, specific symbols are used for initial and final velocities. Understanding what u and v stand for is basic but very important.

Given Data / Assumptions:

• We are dealing with motion along a straight line with constant acceleration.
• The formula given is average velocity = (u + v) / 2.
• Standard notation in kinematics uses u for initial velocity and v for final velocity.
• Displacements are represented by s or x, not by u or v.

Concept / Approach:
In kinematics, u is traditionally used to denote the initial velocity of an object at the start of the time interval considered. The symbol v represents the velocity at the end of that time interval, called the final velocity. When acceleration is constant, the average velocity over the time interval is simply the arithmetic mean of these two velocities, given by (u + v) / 2. Displacement is a separate quantity with different symbols. Therefore, in the given formula, u clearly represents initial velocity.

Step-by-Step Solution:
Step 1: Recall the standard symbols used in equations of motion: u for initial velocity, v for final velocity, a for acceleration, t for time.Step 2: Recognise the formula average velocity = (u + v) / 2 as valid for uniform acceleration.Step 3: Understand that this formula computes the mean of the starting and ending velocities.Step 4: Since v is the final velocity, u must correspond to the starting velocity.Step 5: Note that displacement is not represented by u in any standard equations; displacement uses s or x.Step 6: Conclude that in this formula, u represents the initial velocity.

Verification / Alternative check:
Consider a motion starting with velocity u and ending with velocity v after time t under constant acceleration a. Other equations of motion are v = u + a * t and s = u * t + 0.5 * a * t^2. These equations all treat u as the velocity at time t = 0. When plotting velocity against time, the graph is a straight line from u to v, and the average of the two endpoints is (u + v) / 2. This reinforces that u is initial velocity and fits consistently with all standard kinematic relations.

Why Other Options Are Wrong:
Final velocity is represented by v, not u, in standard notation. Initial displacement and final displacement are not represented by u and are usually written as x0 and x or s. Average velocity is calculated using displacements and time, but the symbol u itself refers to velocity, not displacement. Therefore, interpreting u as any type of displacement is incorrect.

Common Pitfalls:
Students occasionally forget the standard symbols and may mix up u and v, especially if they are new to kinematics. Others might think that u is a generic symbol and assign it different meanings in different questions. To avoid confusion, always remember the standard convention: u is initial velocity, v is final velocity, and these definitions remain consistent across all basic equations of motion.

Final Answer:
In the formula average velocity = (u + v) / 2, u represents the initial velocity.