Vertical velocity to reach a certain point given a flat horizontal velocity? [closed]

Suppose I have two points such as $a(4,6,9)$ and $b(32,5,12)$.

If I have a flat velocity pointing $a$ to $b (b - a)$ which has an arbitrarily defined magnitude. Given gravity $g$,how can I calculate the horizontal upwards velocity required to ensure that an object can be propelled from point $a$ and land on point $b$?

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closed as too localized by David Z♦Feb 2 '13 at 23:59

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Hi meds, and welcome to Physics Stack Exchange! This is a site for conceptual questions about physics, not general homework help. If you can edit your question to show what you've tried, why it didn't work, and ask about the specific physics concept that is giving you trouble, I'll be happy to reopen it. See our FAQ and homework policy for more information. –  David Z Feb 3 '13 at 0:00

Let us assume that the horizontal direction is given by the coordinate $x$ and the vertical direction by $y$. We want to make use of the formula for the trajectory of a point particle in a gravitational field, which can be acquired by integrating acceleration twice, and is given by

$\vec{r}=\frac{1}{2}\vec{g}t^2+\vec{v_0}t+\vec{r_0},$

where $\vec{r}$ is the trajectory, $\vec{g}$ the gravitational acceleration, $\vec{v_0}$ the initial velocity, $\vec{r_0}$ the initial position and $t$ time. We need to treat the components separately, for which we get

$r_x=v_{0x}t+r_{0x}$

and

$r_y=-\frac{1}{2}gt^2+v_{0y}t+r_{0y}.$

In these equations, we know all quantities except for $t$ and $v_{0y}.$ Since we have two equations for two unknown variables, the problem can easily be solved. All you have to do is express $t$ from the first equation and plug it into the second equation. Then you can find an expression for your desired quantity, the vertical velocity.

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