From a point on the ground a particle is projected with initial velocity u, such that its horizontal range is maximum. The magnitude of average velocity during its ascent is

Explanation

Velocity of projection = u .
Range is maximum `rArr theta = 45^(@)`
During time of ascent `rArr ` when h = `h_(mas)`
`rArr u_(x) = u _(x) = u * cos theta`
Average velocity, `V_(A) = sqrt(V_(x)^(2) + V_(y)^(2))`
`V_(x)` = Average velocity along x-axis
` = (u* cos theta + u cos theta)/(2)`
`v_(x) = u * cos theta = u * 45^(@) = u //sqrt(2) to (1)`
Average velocity along y-axis `= (u_(y) + v_(y))/(2)`
`u_(y) = u sin theta = u // sqrt(2), v_(y) = 0 (at h_(max))`
`:. v_(y) = (u//sqrt(2))/(2) = (u)/(2sqrt(2)) rarr (2)` Average velocity during time of ascent
`sqrt((u/(sqrt(2)))^(2)+((u)/(2sqrt(2)))^(2))`
Average velocity
`= sqrt(u^(2)/(2)+(u^(2))/(4 xx 2))=sqrt((4u^(2) + u^(2))/(8))=(usqrt(5))/(2sqrt(2))`