A body is released from rest at `t_(0) = 0, x_(0) = 0` and moves with constant acceleration. Its position is measured at the equally spaced times `t_(1) = Delta t, t_(2) = 2 Delta t, t_(3) = 3 Delta t` and so forth. Prove that displacements `Delta x_(j) - x_(j - 1)` which occur in the successive time intervals `Delta t` are in the ratio
`Delta x_(1) : Delta x_(2) : Delta x_(3)...... ~~ 1 : 3 : 5...`

Explanation

`{:(t_(1) = 1",",x_(1) = (1)/(2) xx g xx t_(1)^(2)",",x_(1) = 5 m",",Delta x_(1) = 5,),(t_(2) = 2",",x_(2) = (1)/(2) xx g xx t_(2)^(2)",",x_(2) = 20 m,Delta x_(2) = 15,),(t_(3) = 3",",x_(3) = (1)/(2) xx g xx t_(3)^(2)",",x_(3) = 45 m",",Delta x_(3) = 25,):}`