91Ó°ÊÓ

Average velocity in the context of uniform acceleration is an essential concept to grasp, especially when faced with problems involving motion in physics. When an object moves with uniform acceleration, its velocity changes at a constant rate. Average velocity describes the object's velocity over a given time interval. It is the sum of the initial velocity and final velocity, divided by two. In mathematical terms:
\[ v = \frac{u + u + at}{2} = u + \frac{at}{2} \]

• Here, \(u\) represents the initial velocity.
• \(a\) is the acceleration, which is constant.
• \(t\) is the time duration for which the object has been accelerating.

For three successive intervals, these average velocities become:

• \( v_1 = u + \frac{a t_1}{2} \)
• \( v_2 = u + \frac{a (t_1 + t_2)}{2} \)
• \( v_3 = u + \frac{a (t_1 + t_2 + t_3)}{2} \)

Each of these equations helps in understanding how the average velocity changes over different intervals of motion.

When assessing changes in velocity over time intervals, the differences between the average velocities for successive intervals come into play. Velocity difference tells us how much the average velocity has changed from one interval to the next, indicating acceleration's role in the object's movement. To calculate the differences:
For the first interval into the second:

• \( v_1 - v_2 = \left( u + \frac{a t_1}{2} \right) - \left( u + \frac{a (t_1 + t_2)}{2} \right) = -\frac{a t_2}{2} \)

For the second interval into the third:

• \( v_2 - v_3 = \left( u + \frac{a (t_1 + t_2)}{2} \right) - \left( u + \frac{a (t_1 + t_2 + t_3)}{2} \right) = -\frac{a t_3}{2} \)

Both the differences are negative, emphasizing that the velocities decrease through each stage—an effect of the consistent uniform acceleration. These values shed light on how each time period contributes to the total change in motion.

Time intervals play a crucial role when evaluating motion under uniform acceleration. Each interval, marked by \( t_1, t_2, \) and \( t_3 \), signifies a separate part of the overall journey, contributing to changes in velocity as seen in the earlier sections.
Uniform acceleration means the rate of change of velocity remains constant across these intervals. This constancy allows us to relate velocity differences directly to their respective time intervals. As we discovered from calculating velocity differences:

• In the ratio \( \frac{v_1 - v_2}{v_2 - v_3} = \frac{t_2}{t_3} \), the velocity differences are directly proportional to the durations of the intervals that define those differences.

Understanding how time intervals impact velocity change helps in forming a more detailed perspective on an object's movement. Each interval is a piece of the larger motion puzzle, and knowing the relationships between velocity differences and time intervals is key to deciphering motion under uniform acceleration.