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

See figure, where C is the middle point of AB. Let v and a be the velocity and acceleration respectively at point C. ThenFor A to B, β2=α2+2ad....(i)For A to C v2=α2+2a(d/2)....(2)From eq. (ii), 2v2=2α2+2ad=α2+α2+2ad=α2+β2∴v2=α2+β22or v=α2+β22