Determine the time after which the velocity of a ball,projected with speed v0 at an angle a0 from a point, becomes perpendicular to the velocity of...

We'll choose the time t = 0 when the ball is projected.
After the time t, it's velocity vector becomes perpendicular to the velocity vector of
projection v0.

We'll write the law that describes the
velocity of the ball at time t:

v = v0 + g*t
(1)

If v is perpendicular to v0, then the dot product of
the vectors is:

v*v0 = 0

the
formula for dot product is:

v*v0 = |v|*|v0|*cos
(v,v0)

The angle between v and v0 is 90 degrees, so cos 90
= 0.

v*v0 = |v|*|v0|*0

v*v0 =
0

 We'll substitute v by the law
(1):

(v0 + g*t)*v0 =
0

We'llremove the
brackets:

v0^2 + g*t*v0 = 0
(2)

We'll substitute v0 = v0*cos(pi/2 + a0)
(3)

We'll substitute (3) in
(2):

v0^2 + g*t*v0*cos(pi/2 + a0) =
0

But cos(pi/2 + a0) = sin
a0

v0^2 + g*t*v0*sin a0 =
0

We'll factorize by v0:

v0(v0
+ g*t*sin a0) = 0

If v0 is different of zero, then the
other factor must be zero:

v0 + g*t*sin a0 =
0

We'll subtract v0:

g*t*sin
a0 = -v0

We'll divide by g*sin
a0:

t = - v0/g*sin
a0

The velocity of the ball
becomes perpendicular to the velocity of projection after the time t = - v0/g*sin
a0.