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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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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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