How to calculate this simple animation effect (physics engine)?

Question

I am implementing a very simple animation effect for a game. The scenario is like this:

1. there is a elastic rubber line, length is 1 meter, when it is extended over 1 meter, it is elastic.

2. the line connects two dots A and B like this, the distance is S, S > 1 meter

A <------------- B

3. then fix dot A, and releases B, the line takes B to the direction of A

I want to know how to calculate time T, which B costs to move X meters towards A (X <= S).

Any ideas? Thanks!

Answer 1

I have been meaning to learn how to animate these kinds of images in sage (a python based platform for math) for a while, so i used this as an excuse. I hope this code snippet and image is helpful.

A = 3
w = 0.5 

# x = f(t) = A cos(wt) inside elastic region
# with x = displacement from 1 meter mark
# in the below code, x is the displacement from origin (x = A cos(wt) + 1)

# find speed when we cross the one meter mark
# f'(t) = -Aw sin(wt), but this is also max speed
# ie f'(t at one meter mark) = -Aw

speed_max = -A * w

# time to reach max speed + time to cross last meter
eta  = float(pi/2 * 1/w + 1/abs(speed_max))

# the function you were looking for
def time_left(x):
    if x < 1:
        return x/abs(speed_max)
    else:
        return 1/w * arccos((x-1)/A)

It may not be clear in the image but within one meter of the origin there is no acceleration.