A particle moves with constant acceleration. | vedclass

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(A) Let $u_1, u_2, u_3, u_4$ be the velocities at $t=0, t_1, (t_1+t_2)$ and $(t_1+t_2+t_3)$ respectively. Let $a$ be the constant acceleration.The ave

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

Vedclass · Answer

(A) Let $u_1, u_2, u_3, u_4$ be the velocities at $t=0, t_1, (t_1+t_2)$ and $(t_1+t_2+t_3)$ respectively. Let $a$ be the constant acceleration.The average velocity in an interval is the arithmetic mean of initial and final velocities: $v_1 = \frac{u_1 + u_2}{2}$,$v_2 = \frac{u_2 + u_3}{2}$,$v_3 = \frac{u_3 + u_4}{2}$.Using $v = u + at$:$u_2 = u_1 + at_1$$u_3 = u_2 + at_2 = u_1 + a(t_1 + t_2)$$u_4 = u_3 + at_3 = u_1 + a(t_1 + t_2 + t_3)$Now,$v_1 - v_2 = \frac{u_1 + u_2}{2} - \frac{u_2 + u_3}{2} = \frac{u_1 - u_3}{2} = \frac{u_1 - (u_1 + a(t_1 + t_2))}{2} = -\frac{a(t_1 + t_2)}{2}$.Similarly,$v_2 - v_3 = \frac{u_2 + u_3}{2} - \frac{u_3 + u_4}{2} = \frac{u_2 - u_4}{2} = \frac{(u_1 + at_1) - (u_1 + a(t_1 + t_2 + t_3))}{2} = -\frac{a(t_2 + t_3)}{2}$.Dividing the two expressions:$\frac{v_1 - v_2}{v_2 - v_3} = \frac{t_1 + t_2}{t_2 + t_3}$.

$A$ particle moves with constant acceleration. Let $v_1, v_2, v_3$ be the average velocities in successive time intervals $t_1, t_2$ and $t_3$. Then:

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