Re: Mathematical model of inertia


"MobyDikc" <mobydikc@xxxxxxxxx> wrote in message news:1140447956.172319.119380@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
MobyDikc wrote:
Can someone provide a mathematical model of inertia?


Let's try this.

Can someone provide the mathematical statement of "unless acted on by
an outside force an object in motion stays in motion and an object at
rest stays at rest."?


Next, can someone provide the work to use that math to answer the
following question.

There is a ball with a mass of 20kg moving toward a 5kg ball at 1 m/s.
At 0 seconds, the balls are 2 meters apart.

What is the state of the system at 3 seconds.

Show all the math involved to make that prediction.

Before collision:
World line of ball_1: x = v t
World line of ball_2: x = d
Total momentum: m1 v
Total energy: 1/2 m1 v^2

This setup satisfies the initial conditions that at t = 0
the distance between the balls is d.

Collision event:
{ x = v t
{ x = d
giving
( x, t ) = ( d, d/v )

After collision:
World line of ball_1: x - d = v1 (t-d/v)
World line of ball_2: x - d = v2 (t-d/v)
Total momentum: m1 v1 + m2 v2
Total energy: 1/2 m1 v1^2 + 1/2 m2 v2^2

Assuming total inelasticity of the collision,
conservation of momentum demands
m1 v = m1 v1 + m2 v2
and conservation of energy demands
1/2 m1 v^2 = 1/2 m1 v1^2 + 1/2 m2 v2^2

Solving for v1 and v2 gives
v1 = (m1-m2) / (m1+m2)
v2 = 2 m1 v / (m1+m2)

So we have the complete histories for both balls:
1) for t < d/v:
World line of ball_1: x = v t
World line of ball_2: x = d
2) for t >= d/v:
World line of ball_1: x = d + (m1-m2) / (m1+m2) (t-d/v)
World line of ball_2: x = d + 2 m1 v / (m1+m2) (t-d/v)

Your question
| > There is a ball with a mass of 20kg moving toward a 5kg ball at 1 m/s.
| > At 0 seconds, the balls are 2 meters apart.
| > What is the state of the system at 3 seconds.
is translated into
m1 = 20, m2 = 4, v = 1, d = 2, T = 3

You can make the calculations.

Dirk Vdm


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