Example: "Combat--Flying Rocks" video

"Combat: Flying Rocks"
RSMilitaryVids
youtu.be/Ix3GU8wLuTo

In this video dating back to at least 2006, flying rocks from an explosion nearly strike U.S. soldiers awaiting atop a nearby building.

Let's assume the following quantities as givens:

x₀ = 0 m,
t₀ = 0 s,
a_y = –9.80 m/s².

What else that can be determined from the video? Software analysis of the video reveals a rate 24 frames per second, and 40 frames elapse from the first flash of the explosion, to when the rock hits the wall near the soliders' heads, such that the elapsed time can be calculated:

t = 40 frames·((1 sec)/(24 frames)) = 1.7 s.

Also we can make a reasonable simplification that the rock hits the wall at the highest point of its trajectory, such that the final vertical velocity component is zero:

v_y = 0 m/s.

(If this seems like a drastic oversimplification, consider that we are already neglecting drag in an idealized model of projectile motion.)

The final horizontal velocity component of the rock can then be estimated from two adjacent frames, from hitting the wall, to moving approximately 2.0 m horizontally (the width of two soldiers standing shoulder-to-shoulder) after 1 frame:

v_x = (2.0 m/frame)·((24 frames)/(1 sec)) = +48 m/s.

And since we are neglecting drag, then the initial horizontal velocity vector v_0x = +48 m/s as well.

So in the equations for projectile motion, the following quantities are still unknown:

x = v_0x·t,

v_y = v_0y + a_y·t,

y = (1/2)·(v_y + v_0y)·t,

y = v_0y·t + (1/2)·a_y·t²,

v_y² – v_0y² = 2·a_y·y,

Note that there are only three unknowns (x, v_0y, and y) in these five equations, so at this point the problem is sufficiently determined. Now in the first equation we can immediately solve for the final horizontal position of the flying rock as it hits the wall:

x = v_0x·t = (+48 m/s)·(1.7 s) = +83 m.

From the second equation, the initial vertical velocity component can now be solved for:

v_0y = v_y – a_y·t = (0 m/s) – (–9.80 m/s²)·(1.7 s) = +17 m/s.

Then the final unknown quantity, the final vertical position of the flying rock as it hits the wall, can be solved for from the third equation:

y = (1/2)·(v_y + v_0y)·t = (1/2)·((0 m/s) + (+17 m/s))·(1.7 s) = +14 m.