Re: physical significance
From: Alex Hunsley (lardattardisdoteddotacdotuk_at_mailinator.com)
Date: 12/14/04
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Date: Tue, 14 Dec 2004 12:21:43 +0000
shashidhar wrote:
> Hello All,
>
> I am a beginner in mathematics and i am stuck in the following concept i
> found in a book of complex numbers:
>
>
>
> "Consider a child throwing a ball into the air.
>
> For example, assume that the ball is thrown straight up, with an initial
>
> velocity of 9.8 meters per second. One second after it leaves the child's
>
> hand, the ball has reached a height of 4.9 meters, and the acceleration of
>
> gravity (9.8 meters per second2) has reduced its velocity to zero. The ball
>
> then accelerates toward the ground, being caught by the child two seconds
>
> after it was thrown. From basic physics equations, the height of the ball at
>
> any instant of time is given by:
>
>
>
> h = (-g*t^2)/2 + v*t
>
>
>
> where h is the height above the ground (in meters), g is the acceleration of
>
> gravity (9.8 meters per second2), v is the initial velocity (9.8 meters per
>
> second), and t is the time (in seconds).
>
> t ' 1± 1&h/4.9
>
> Now, suppose we want to know when the ball passes a certain height.
>
> Plugging in the known values and solving for t:
>
> For instance, the ball is at a height of 3 meters twice: t =0.38 (going up)
>
> and t = 1.62 seconds (going down).
>
> As long as we ask reasonable questions, these equations give reasonable
>
> answers. But what happens when we ask unreasonable questions? For
>
> example: At what time does the ball reach a height of 10 meters? This
>
> question has no answer in reality because the ball never reaches this
> height.
>
> Nevertheless, plugging the value of h = 10 into the above equation gives two
>
> answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041)."
>
>
>
>
>
> My question is, in the above example what is the Physical significance of
> the complex time?
>
> I believe this quantity of complex time would be used in some physical
> concept or theory....How do we analyse this complex time in the real world?
I'm not sure of the physical significance of complex time per se, but in
the case of this problem (and other quadratic based problem solutions I
suspect), I think the complex answer does have a meaningful
interpretation of a sort.
For a start, the real part of the answer represents the time at which
the object got *closest* to the target distance (10, in your example).
In the complex answer case, at this time the object has velocity zero.
As for the imaginary part....
Hint: write out the kinematic equation h = (-g*t^2)/2 + v*t in its
purely quadratic form (in terms of t), then write out the expression for
the solution if this quadratic. (You know; -b +/- ...etc) Identify which
part of this expression gives the imaginary term. Then look at the
equation for this imaginary term and compare it to the standard 5 laws
of kinematics and see what it's similar to.
alex
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