I received this question in my recent homework and don't know how to approach it: "A particle is projected from a height of 30m above the ground, with initial velocity 3i+4j. Find the time it takes for a particle to be traveling perpendicular to original projection"
Any help would be appreciated.
Observe that gravity does not affect horizontal velocity, so that when the conditions of the question are satisfied, the projectile velocity will be $3i+aj$, where $a$ needs to be determined. The easiest way to do so is to note that the dot product of two perpendicular non-zero vectors is zero; in this case, it tells us that
$$ (3, 4) \cdot (3, a) = 3 \times 3 + 4 \times a = 0 $$
Simple algebra tells us what $a$ is, and then the time it takes to do so is given by
$$ t = \frac{4-a}{9.8} $$
where the $4$ comes from the initial vertical component of velocity. The only relevance of the $30$ m initial height that I can see is the determination of whether the projectile ever travels perpendicular to its original vector, or if it hits the ground first. But in this case, I think you should be able to verify, fairly easily, that it does not hit the ground first (provided the ground is level).