A radius vector of a point A relative to the origin varies with time t as `r=ati-bt^2j`, where a and b are positive constants, and i and j are the unit vectors of the x and y axes. Find:
(a) the equation of the point's trajectory `y(x)`, plot this function,
(b) the time dependence of the velocity v and acceleration w vectors, as well as of the moduli of these quantities,
(c) the time dependence of the angle `alpha` between the vectors w and v,
(d) the mean velocity vector averaged over the first t seconds of motion, and the modulus of this vector.

Explanation

`vec( r) = alpha t hat(i) - beta t^(2) hat(j) = xhat(i) + y hat(j)`
(a) `x = alpha t rArr t = (x)/(alpha)`
`y = - beta t^(2) = - beta(x^(2))/(alpha^(2))`
`y = - (beta x^(2))/(alpha^(2))` (parabola, open downward)
(b) `vec( r) = alpha t hat(i) - beta t^(2) hat(j)`
`vec(v) (d vec(r))/(dt) = alpha hat(i) - 2 beta - 2 beta t hat(j)`
`vec(a) = (d vec(v))/(dt) = 0 2beta hat(j)`
`|vec(v)| = v = sqrt((alpha)^(2) + (- 2 beta)^(2)) = sqrt(alpha^(2) + 4 beta^(2) t^(2))`
`|vec(a)| = 2 beta`
( c) For the angle between `vec(v)` and `vec(a)`
`vec(v).vec(a) = 4 beta^(2) t`
`cos theta = (vec(v).vec(a))/(|vec(v)||vec(a)|) = (4 beta^(2) t)/(sqrt(alpha^(2) + 4 beta^(2) t^(2) (2 beta))`
`= (2 beta t)/(sqrt(alpha^(2) + 4beta^(2) t^(2))`
where `theta`is the angle between `vec(v)` and `vec(a)`.
(d) `vec(v) = (int_(0)^(t)vec(v) dt)/(int_(0)^(t) dt) = (int_(0)^(t)(alpha hat(i) - 2 beta t hat(j)) dt)/(t)`
`= (alpha t hat(i) - 2 beta.(t^(2))/(2)hat(j))/(t)`
`= alpha hat(i) - beta t hat(j)`
`|bar vec(v)| = sqrt(alpha^(2) + beta^(2) t^(2))`