Understanding the Concept of Average Velocity in Uniformly Accelerated Motion

The formula (frac{u v}{2}) represents the average velocity ({v}_{text{av}}) of an object moving in a straight line with uniform acceleration. In this context, (u) denotes the initial velocity, and (v) represents the final velocity of the object.

Concept Explanation

Uniform Acceleration

The formula assumes that the object is experiencing uniform acceleration, meaning its acceleration remains constant over the time interval considered. This constant acceleration allows for a straightforward calculation of the average velocity.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. When an object undergoes uniform acceleration, the average velocity can be calculated as the arithmetic mean of the initial and final velocities.

Derivation

Conceptual Explanation

If an object initiates its motion with an initial velocity (u) and reaches a final velocity (v) after a certain time (t), the average velocity over that time interval can be thought of as the midpoint between these two velocities. Mathematically, when an object moves in a straight line with constant acceleration, the average velocity can be directly calculated as the average of the initial and final velocities.

Calculus-Based Derivation

The most general equation for the average velocity over a time interval (T) is given by:

[langle v rangle frac{1}{T} int_0^T vt dt]

This equation requires a basic understanding of calculus, but the key idea is that it accounts for the possibility of non-uniform acceleration. If the acceleration is constant, the velocity at any time (t) is given by:

[v_t v_0 at]

Where (v_0) is the initial velocity, (a) is the constant acceleration, and (t) is the time elapsed.

The final velocity at time (T) is:

[v_T v u aT]

Substituting this into the equation for (langle v rangle), we get:

[langle v rangle frac{1}{T} int_0^T (u at) dt]

Expanding the integral:

[langle v rangle frac{1}{T} left( uT int_0^T at dt right)]

Evaluating the integral:

[langle v rangle frac{1}{T} left( uT frac{1}{2}aT^2 right)]

Simplifying, we find:

[langle v rangle u frac{1}{2}aT]

Given that (v u aT), we can substitute (aT) with (v - u), leading to:

[langle v rangle frac{u v}{2}]

This shows that the time-averaged velocity in the case of constant acceleration is simply the arithmetic mean of the initial and final velocities, which makes intuitive sense given the linear distribution of velocity over time.

Application

This formula is particularly useful in physics problems involving linear motion, such as calculating the distance traveled over a period of time when the initial and final velocities are known. It simplifies the analysis of motion in scenarios where only these two velocity values are provided.

Conclusion

The formula (frac{u v}{2}) for average velocity is a straightforward and practical tool for solving problems involving uniformly accelerated motion. It highlights the importance of understanding both uniform and non-uniform acceleration and the integration of calculus in more complex scenarios.