First slide

Rectilinear Motion

Question

A point moves with uniform acceleration and v₁, v₂, and v₃ denote the average velocities in the three successive intervals of time t₁, t₂, and t₃ Which of the following relations is correct?

Difficult

A

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{2} + t_{3})$
B

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} + t_{2}) : (t_{2} + t_{3})$
C

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{1} - t_{3})$
D

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{2} - t_{3})$

Solution

Suppose u be the initial velocity.
$Velocity after time t_{1} : v_{11} = u + a t_{1}$
$Velocity after time t_{1} + t_{2} : v_{22} = u + a (t_{1} + t_{2})$
$Velocity after time t_{1} + t_{2} + t_{3} :$
$\begin{array}{l} v_{33} = u + a (t_{1} + t_{2} + t_{3}) \\ Now v_{1} = \frac{u + v_{11}}{2} = \frac{u + u + a t_{1}}{2} = u + \frac{1}{2} a t_{1} \\ v_{2} = \frac{v_{11} + v_{22}}{2} = u + a t_{1} + \frac{1}{2} a t_{2} \\ v_{3} = \frac{v_{22} + v_{33}}{2} = u + a t_{1} + a t_{2} + \frac{1}{2} a t_{3} \end{array}$
$\begin{array}{l} So v_{1} - v_{2} = - \frac{1}{2} a (t_{1} + t_{2}) \\ v_{2} - v_{3} = - \frac{1}{2} a (t_{2} + t_{3}) \\ (v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} + t_{2}) : (t_{2} + t_{3}) \end{array}$

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