solving t when $t=frac{2*x_{dist}}{v_0*(1+(1-frac{a_{drag%}}{100})^t)}$
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I'm trying to make a function for the time a projectile is traveling over a certain distance, but while trying to solve t I noticed that t is both a power and a base, so use of logarithms would be basically useless. In the case of: $$t=frac{x_{dist}*cosθ}{frac{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}{2}}$$ I went to the form: $$t=frac{2*x_{dist}*cosθ}{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}$$ Which can then be simplified to being: $$t=frac{2*x_{dist}}{v_0*(1+(1-frac{a_{drag%}}{100})^t)}$$ But even after doing these simplifications of the formula you keep the base t and power t, which can't be solved using common logarithmic operations. I tried applying a lot of them as well as looking for other solutions like infinite powers but none seem to get me to a function where t isn't required to calculate itself. Which leads me to think that using limits is the way to solve this equation, because from programming experience I know that repeating this calculation over and over will in the end converge to a certain value
since there is a time this object takes to travel to the destination.
If you are wondering how I made this equation, its made to function within a game to predict the arc of a projectile as accurate as possible. Since the games physics are different from real life (like drag being a percentage of speed per second) I ended up with this formula. In the end I would want to know the angle to shoot at, but that isn't relevant for this question since I could simplify the equation to not involve the angle anymore.
So now the real question:
How would I, given this formula, make a function for t and thus get rid of the power in $(1-frac{a_{drag%}}{100})^t$?
I ploted the graph in desmos to verify that it actually converged and got the following results
calculus multivariable-calculus
asked Nov 18 at 9:56
Timo Post
11
add a comment |
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0
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I'm trying to make a function for the time a projectile is traveling over a certain distance, but while trying to solve t I noticed that t is both a power and a base, so use of logarithms would be basically useless. In the case of: $$t=frac{x_{dist}*cosθ}{frac{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}{2}}$$ I went to the form: $$t=frac{2*x_{dist}*cosθ}{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}$$ Which can then be simplified to being: $$t=frac{2*x_{dist}}{v_0*(1+(1-frac{a_{drag%}}{100})^t)}$$ But even after doing these simplifications of the formula you keep the base t and power t, which can't be solved using common logarithmic operations. I tried applying a lot of them as well as looking for other solutions like infinite powers but none seem to get me to a function where t isn't required to calculate itself. Which leads me to think that using limits is the way to solve this equation, because from programming experience I know that repeating this calculation over and over will in the end converge to a certain value
since there is a time this object takes to travel to the destination.
If you are wondering how I made this equation, its made to function within a game to predict the arc of a projectile as accurate as possible. Since the games physics are different from real life (like drag being a percentage of speed per second) I ended up with this formula. In the end I would want to know the angle to shoot at, but that isn't relevant for this question since I could simplify the equation to not involve the angle anymore.
So now the real question:
How would I, given this formula, make a function for t and thus get rid of the power in $(1-frac{a_{drag%}}{100})^t$?
I ploted the graph in desmos to verify that it actually converged and got the following results
calculus multivariable-calculus
asked Nov 18 at 9:56
Timo Post
11
You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to make a function for the time a projectile is traveling over a certain distance, but while trying to solve t I noticed that t is both a power and a base, so use of logarithms would be basically useless. In the case of: $$t=frac{x_{dist}*cosθ}{frac{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}{2}}$$ I went to the form: $$t=frac{2*x_{dist}*cosθ}{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}$$ Which can then be simplified to being: $$t=frac{2*x_{dist}}{v_0*(1+(1-frac{a_{drag%}}{100})^t)}$$ But even after doing these simplifications of the formula you keep the base t and power t, which can't be solved using common logarithmic operations. I tried applying a lot of them as well as looking for other solutions like infinite powers but none seem to get me to a function where t isn't required to calculate itself. Which leads me to think that using limits is the way to solve this equation, because from programming experience I know that repeating this calculation over and over will in the end converge to a certain value
since there is a time this object takes to travel to the destination.
If you are wondering how I made this equation, its made to function within a game to predict the arc of a projectile as accurate as possible. Since the games physics are different from real life (like drag being a percentage of speed per second) I ended up with this formula. In the end I would want to know the angle to shoot at, but that isn't relevant for this question since I could simplify the equation to not involve the angle anymore.
So now the real question:
How would I, given this formula, make a function for t and thus get rid of the power in $(1-frac{a_{drag%}}{100})^t$?
I ploted the graph in desmos to verify that it actually converged and got the following results
calculus multivariable-calculus
asked Nov 18 at 9:56
Timo Post
11
I'm trying to make a function for the time a projectile is traveling over a certain distance, but while trying to solve t I noticed that t is both a power and a base, so use of logarithms would be basically useless. In the case of: $$t=frac{x_{dist}*cosθ}{frac{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}{2}}$$ I went to the form: $$t=frac{2*x_{dist}*cosθ}{v_0*cosθ+v_0*cosθ*(1-frac{a_{drag%}}{100})^t}$$ Which can then be simplified to being: $$t=frac{2*x_{dist}}{v_0*(1+(1-frac{a_{drag%}}{100})^t)}$$ But even after doing these simplifications of the formula you keep the base t and power t, which can't be solved using common logarithmic operations. I tried applying a lot of them as well as looking for other solutions like infinite powers but none seem to get me to a function where t isn't required to calculate itself. Which leads me to think that using limits is the way to solve this equation, because from programming experience I know that repeating this calculation over and over will in the end converge to a certain value
since there is a time this object takes to travel to the destination.
If you are wondering how I made this equation, its made to function within a game to predict the arc of a projectile as accurate as possible. Since the games physics are different from real life (like drag being a percentage of speed per second) I ended up with this formula. In the end I would want to know the angle to shoot at, but that isn't relevant for this question since I could simplify the equation to not involve the angle anymore.
So now the real question:
How would I, given this formula, make a function for t and thus get rid of the power in $(1-frac{a_{drag%}}{100})^t$?
I ploted the graph in desmos to verify that it actually converged and got the following results
calculus multivariable-calculus
calculus multivariable-calculus
asked Nov 18 at 9:56
Timo Post
11
asked Nov 18 at 9:56
Timo Post
11
edited Nov 18 at 10:15
asked Nov 18 at 9:56
Timo Post
11
asked Nov 18 at 9:56
Timo Post
11
asked Nov 18 at 9:56
Timo Post
11
11
You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05
add a comment |
You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05
You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05
add a comment |
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You can use Taylor expansion to solve this equation as accurately as you want. I suppose that for a game it will be enough to expand the power to the second or third term
– Mikalai Parshutsich
Nov 18 at 14:42
That's one way of doing it, but I as a (beginning) programmer prefer a more elegant solution. If it's impossible or really hard to solve for t I'll most likely stick to this option though since it's the second best substitute for just getting the right equation.
– Timo Post
Nov 18 at 19:05