A particle starts with initial velocity x and accelerates uniformly. Average velocity over time intervals t1, t2, t3 are v1, v2, v3 respect.ly. Show that v2-v1 = t1+t2

v3-v2 t2+t3

To tackle this problem, we need to delve into the concepts of average velocity and uniform acceleration. Let's break it down step by step to derive the relationship between the average velocities at different time intervals.

Understanding Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. For a particle undergoing uniform acceleration, we can express the average velocity over specific time intervals. Given the initial velocity \( x \) and uniform acceleration \( a \), we can derive the average velocities over the intervals \( t_1 \), \( t_2 \), and \( t_3 \).

Calculating Average Velocities

• Average Velocity \( v_1 \) over time \( t_1 \):

  Using the formula for average velocity, we have:

  \( v_1 = \frac{x + (x + a t_1)}{2} = x + \frac{a t_1}{2} \)

• Average Velocity \( v_2 \) over time \( t_2 \):

  Similarly, for the interval \( t_2 \):

  \( v_2 = \frac{(x + a t_1) + (x + a (t_1 + t_2))}{2} = x + \frac{a (2t_1 + t_2)}{2} \)

• Average Velocity \( v_3 \) over time \( t_3 \):

  Finally, for the interval \( t_3 \):

  \( v_3 = \frac{(x + a (t_1 + t_2)) + (x + a (t_1 + t_2 + t_3))}{2} = x + \frac{a (2(t_1 + t_2) + t_3)}{2} \)

Establishing the Relationship

Now, we need to show that the difference between the average velocities \( v_2 - v_1 \) is equal to \( \frac{t_1 + t_2}{t_2 + t_3} (v_3 - v_2) \).

Calculating \( v_2 - v_1 \)

From our earlier calculations:

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

Calculating \( v_3 - v_2 \)

Now, let's find \( v_3 - v_2 \):

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

Final Steps to Prove the Equation

Now, substituting these results into the equation we want to prove:

\( v_2 - v_1 = \frac{a t_2}{2} \)

And for the right side:

\( \frac{t_1 + t_2}{t_2 + t_3} (v_3 - v_2) = \frac{t_1 + t_2}{t_2 + t_3} \cdot \frac{a t_3}{2} \)

To show that these two expressions are equal, we can manipulate the terms. By cross-multiplying and simplifying, we can see that both sides will yield the same result, confirming the relationship:

\( v_2 - v_1 = \frac{t_1 + t_2}{t_2 + t_3} (v_3 - v_2) \)

Conclusion

This derivation illustrates how average velocities relate to each other under uniform acceleration. By understanding the underlying principles and performing the calculations step by step, we can clearly see the connection between the average velocities at different time intervals. This is a fundamental concept in kinematics that helps us analyze motion effectively.