First slide

The 3 equation for uniform acc

Question

A particle is moving in a straight line with constant acceleration. It crosses two points A and B with velocities $α$ and $β$ respectively. The velocity of particle at mid-point of line AB will be

Moderate

A

$\frac{(α + β)}{2}$
B

$\sqrt{[\frac{(α^{2} + β^{2})}{2}]}$
C

$\frac{(α^{2} + β^{2})}{2 (α + β)}$
D

$\frac{\sqrt{(α^{2} + β^{2})}}{2}$

Solution

See figure, where C is the middle point of AB. Let v and a be the velocity and acceleration respectively at point C. Then
For A to B, $β^{2} = α^{2} + 2 ad . . . . (i)$
For A to C $v^{2} = α^{2} + 2 a (d / 2) . . . . (2)$
From eq. (ii), $2 v^{2} = 2 α^{2} + 2 ad$
$\begin{array}{l} = α^{2} + (α^{2} + 2 a d) \\ = α^{2} + β^{2} \\ ∴ v^{2} = (\frac{α^{2} + β^{2}}{2}) \\ or v = \sqrt{\frac{α^{2} + β^{2}}{2}}] \end{array}$

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