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I'm working on a project which calculates the some statics of a basketball shot. I haven't done physics since high school so I wanted to see if I was on the right track or if I'm completely wrong. Note: this is not a problem for school or nothing like that.

Currently the information I've got to work with is as follows:

  • Height of the hoop
  • Distance from the hoop
  • Height of when the ball was released
  • Time in the air (can be calculated from when it left the players can till it goes in the hoop)

What I don't have (and trying to find):

  • Angle of release
  • Initial velocity

I was following pretty much whats in this video: https://www.youtube.com/watch?v=fNfkYWqB9w8 but since the basketball hoop is a higher elevation that means I have to find that, correct? Could i just use: $y−y_0=(v_yt)−(\frac{1}{2}gt^2)$ where $y =$ height of basketball hoop and $y_0 =$ height of where ball was released? (then solve for $v_y$)

If so I could just use the remaining formulas of $V_x = \Delta x / \Delta t$ and $a^2 + b^2 = c^2$ to find the angle like in the video.

I guess all I'm really asking is to make sure I'm doing this correctly.

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  • $\begingroup$ Where are you getting your data from? Is it a video, by any chance? $\endgroup$ – Philip Jun 29 '20 at 20:17
  • $\begingroup$ Yup, I'm just recording a video and trying to get some information from it like the launch angle. $\endgroup$ – iG Cloud Jun 29 '20 at 20:20
  • $\begingroup$ You should definitely check out Tracker Video Analysis if you haven't already! It's a fantastic tool that should do everything you want, especially if you're the one recording the videos. $\endgroup$ – Philip Jun 29 '20 at 20:22
  • $\begingroup$ Will do, I'm actually trying to create something like that for a learning experience, haha :) $\endgroup$ – iG Cloud Jun 29 '20 at 20:24
  • $\begingroup$ There are probably two solutions for your problem: one where the basketball is shot at a "shallow" angle, and will probably hit the rim, and one where the basketball is shot at a "steep" angle, and will probably "swish" through the hoop. Which solution are you looking for? $\endgroup$ – David White Jun 30 '20 at 1:02
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$\def\th{\theta} \def\ra{\rightarrow}$Suppose the ball is thrown from $(0,h)$ to $(d,H)$ under the influence of gravity in time $t$ and that the initial velocity is ${\bf v}_0 = (v_0\cos\th,v_0\sin\th)$. (In what follows we assume $d>0$ so $-\pi/2<\th<\pi/2$.) We have \begin{align*} d &= v_0 t\cos\th \\ H &= h + v_0 t\sin\th-\frac1 2 g t^2. \end{align*} This is a two-by-two nonlinear system of equations for $(v_0,\th)$. We solve this system with a standard method. The system is equivalent to \begin{align*} v_0 t\cos\th &= d \tag{1}\\ v_0 t\sin\th &= H-h+\frac1 2 g t^2.\tag{2} \end{align*} We square each side of (1) and (2), add, and use Pythagoras' theorem with the result $$v_0^2 t^2 = d^2 + \left( H-h+\frac1 2 g t^2 \right)^2.$$ Thus, $$v_0 = \frac{ \sqrt{d^2 + \left( H-h+\frac1 2 g t^2 \right)}}{t}.$$ If instead we take the ratio of (2) to (1) and solve for $\th$ we find $$\th = \arctan\frac{H-h+\frac1 2 g t^2}{d}.$$ One can check that these results are dimensionally correct and that they "act right" by taking various limits.

(A) For example, suppose $H=h$ and $t\ra 0$. We find $(v_0,\th)\ra(d/t,0)$. If the ball lands at the height from which it was thrown, for short time periods the ball's motion is uniform and horizontal.

(B) If $g\ra 0$ we find $(v_0,\th)\ra(\sqrt{d^2+(H-h)^2}/t,\arctan((H-h)/d))$. That is, if there is no gravitational force, the ball's motion is uniform from $(0,h)$ to $(d,H)$.

(C) If $d\ra 0$ (and $H-h+\frac 1 2 g t^2>0$) we find $(v_0,\th)\ra((H-h+\frac 1 2 g t^2)/t,\pi/2)$. That is, the motion is in the vertical direction and $H=h+v_0 t-\frac 1 2 g t^2$. This is just one-dimensional kinematics of constant accelerated motion.

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Using the Pythagorean theorem on distances is the wrong idea for finding the angle. With the data you have listed , you have a unique solution for $v_{y0}=\text{your }v_y$ and $v_{x0}= \text{your }V_x$. Then find the magnitude and angle of velocity from the components.

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