Frederick Williams wrote:
> Quinch wrote:
>> Just a quick question - I reckon this might be an appropriate
newsgroup,
>> but it's still a shot in the dark since I couldn't find the FAQ.
>>
>> At any rate, I could use some help with a math/physics problem and
would
>> be grateful for any assistance. The question is how to calculate the
>> time needed for an object thrown in the air to fall to a certain point.
>> Thus far all I know for certain is that it's a combination of two
>> acceleration events - the first is the upwards motion, which is, I
>> assume, equal to the time needed for an object to accelerate from a
>> standstill to its initial velocity, and the downward motion, equal to
>> the time needed for the object to accelerate from the apex of the
rising
>> stage to the "ground". I've been looking around, but for the most part,
>> plugging the numbers into the formulas I found got me calculating
square
>> roots of negative numbers, with all the nonsense that implies.
>>
>> So, given the initial height X, the stopping height Y, the initial
>> upward velocity A and the gravity deceleration/acceleration B, how
would
>> I go about calculating the time needed for the arc to complete?
>
> Here are some well known formulae that are useful in problems like this:
>
> u = initial velocity,
> v = velocity after time t,
> s = distance travelled in time t,
> a = acceleration, assumed constant:
>
> v = u + at
>
> s = ut + at^2/2 (*)
>
> v^2 = u^2 + 2as.
>
> The first two follow from the definitions of acceleration being the rate
> of change of velocity, and velocity being the rate of change of
> distance. The third follows from the first two by simple algebra.
>
> In your case, if "up" is positive, then
>
> s = 2Y - X,
>
> u = A (arrow is shot up),
>
> a = -B (gravity pulls down).
>
> (*) will give t in terms of s, u and a:
>
> 2Y - X = At - Bt^2/2
>
> whence
>
> t = (A +/- sqrt(A^2 - 2B(2Y - X)))/B.
>
> What specific numbers were giving you problems?
Thanks for the response {and forgive the lateness of mine}.
The problem with the formula given seems to be in the root - for
example, if we enter constants
X = 50
Y = 50
A = 1
B = 0.1
for the root value we get
Sqrt(1-0.2*(100-50))
Sqrt(1-10)
Sqrt(-9)
....which is completely irrational.
If we try to separate the arc into the upward and downward parts {will
need it for error-checking and maybe some later uses too} can you tell
me if I'm going the right way with this;
Going with the v = u + at equation, we get that
t = v/a
Which basically gives us
Uptime = Speed/gravity
Taking this from the direction of "accelerating from dead stop to
initial speed", I take
s = ut + at^2/2 >>> no initial speed gives us
Apex = Gravity*Uptime^2/2
We can reduce the starting and ending heights to a single variable
simply by subtracting one from the other - afterall, an object will take
the same time to arc between points 5 and 0 as 105 and 100, right?
So that gives us the falling distance of
Fall = Startheight + Apex - Endheight
So given
v^2 = u^2 + 2as.
and
v = at
We get that
(Gravity*Time)^2 = 2*Gravity*Fall
Any suggestions on where I should go with this?
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