Question

A projectile is fired at an angle 53° with horizontal. Find the time after which it makes an angle 45° with horizontal

Answer 1

Fist I'll make the assumption that the angle is positive compared to horizontal

There are 2 situations where the angle is 45 deg.

For the angle to be 45 deg the horizontal velocity EQUALS the vertical velocity

If the initial velocity is Vo then horizontal velocity = Vo cos(53)

The vertical velocity is Vo sin(53) t - g(t^2)/2

SO for those velocities ot be equal

Vocos(53)- Vosin(53)t +g(t^2)/2 = 0

It seems to me that this equation is not independent of Vo and hence cannot be solved unless you are told the initial velocity. I'd be happy to be shown wrong on this - but it seems intuitive that a very high Vo would take longer to reach the 45 deg angle than a lower initial velocity (and it can be seen that a Vo approaching 0 will result in a very SHORT time to reach the 45 deg point.

I think you need Vo to solve this.