Free fall with marks and endpoint
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates free fall motion in real time, neglecting air resistance. The animation does not use formulas (no equations or differential calculus), but only makes the necessary variations in the vectors that direct the motion.
This animation follows the same steps as the previous free fall simulation, but now we add marks every second of the fall (see details below).
As you can see, the results match reality quite well. The green marks represent the height reached by the blue point at each second according to the animation. The orange marks indicate the theoretical height that the point should be at after each second.
Also, note that if there is no friction, in each passing second, the mass always covers the same height, regardless of the initial height (4.905 is half the constant value of |g|, 9.81 m/s²):
After 1 second, it has fallen 4.905 m for 12 (that is, 4.905 m).
After 2 seconds, it has fallen 4.905 m for 22 (that is, 19.62 m).
After 3 seconds, it has fallen 4.905 m for 32 (that is, 44.145 m).
After 4 seconds, it has fallen 4.905 m for 42 (that is, 78.48 m).
...
- Note: There is an intuitive way (without resorting to differential calculus) to understand this. By definition of acceleration, we know that after t seconds, the velocity v is g t. But this is the final velocity, while the initial was 0. Since acceleration is constant, that is, equal at any instant, it is reasonable to think that the mass will take the same time as if it carried the average velocity of both: (0 + g t)/2 = g/2 t (this general result is known as the mean velocity theorem). Therefore, the distance traveled will be the module of this velocity multiplied by the elapsed time: |g|/2 t2 = 4.905 t2. In free fall from height h, we know that v = g t and we know that h = |g|/2 t2, by solving for t in the second equation and substituting in the first, we find that the magnitude of that velocity is always
After recording the time and executing the movement as shown in the previous activity, to register the marks, we add the variable:
last = 0
and the lists:
reg = {0}
mark = {}
Now, we just need to add the following script to the slider anima:
SetValue(last, reg(1))
SetValue(reg, If(floor(t)>last, Append(floor(t), reg), reg))
SetValue(mark, If(floor(t)>last, Append(y(M), mark), mark))
The sequence of theoretical marks (in orange) is:
marksT = Sequence(Segment((x(P), y(P) - 4.905k²), (0, y(P) - 4.905k²)), k, 1, Length(mark))
The sequence of animation marks (in green) is:
marks = Sequence(Segment((x(P), mark(k)), (x(P) + x(Corner(2) - Corner(1)) / 20, mark(k))), k, 1, Length(mark))
And the sequence for the seconds is:
marksN = Sequence(Text("" reg(k) + "''", (x(P) + x(Corner(2) - Corner(1)) / 18, mark(k))), k, 1, Length(mark))
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Register passing through an integer number of seconds and the corresponding height
SetValue(last, reg(1))
SetValue(reg, If(floor(t) > last, Append(floor(t), reg), reg))
SetValue(mark, If(floor(t) > last, Append(y(M), mark), mark))
# Move M and control the endpoint
SetValue(v, v + dt g)
SetValue(M, M + dt v)
StartAnimation(anima, y(M) > 0)
Author of the activity and GeoGebra construction: Rafael Losada.