Let's imagine a bird flying with a constant horizontal velocity, , at a height, . It drops a projectile which is then only acted upon by gravity, .
Our goal is to define a "probability of impact," . A simple way to think about this is to consider the alignment of the projectile's final velocity vector, , with the horizontal ground vector, . The more aligned they are (a shallower angle), the "better" the shot, in a sense.
The initial idea was to use a dot product, but a much more elegant solution is to normalize it. This gives us the cosine of the impact angle, .
To find , we first need to find the angle itself. This requires calculating the projectile's final velocity components at the moment of impact.
Horizontal Velocity ( ): Assuming no air resistance, the horizontal velocity remains constant.
Time of Flight ( ): The time it takes to fall a height from rest is found using:
Final Vertical Velocity ( ): The vertical speed upon impact is:
Impact Angle ( ): The angle is the arctangent of the ratio of the velocity components.
Now we have our "probability" defined as . This can be greatly simplified using a right-angled triangle.
If , we can construct the following triangle:
From this, we can see that . This gives us our final, simplified formula:
This powerful formula tells us exactly how each variable affects our impact "probability":
In conclusion, the highest "probability" in this model is achieved when the horizontal velocity is far greater than the final vertical velocity, leading to a shallow, grazing impact.