By applying algebraic handling, exponential and logarithm properties, we conclude that the equation of the time of flight (T) is [tex]T = -\frac{1}{k}\cdot \ln \frac{\frac{g}{k} }{v_{o}+\frac{g}{k} }[/tex].
How to clear a variable within an exponential equation
In this question we must clear the variable of time of flight (T) by means of algebraic procedures. Exponential expressions are trascendental functions, these are, expressions that cannot be described algebraically:
[tex]v_{o}\cdot e^{-k\cdot T} - \frac{g}{k} \cdot (1 - e^{-k\cdot T}) = 0[/tex]
[tex]\left(v_{o} + \frac{g}{k} \right)\cdot e^{-k \cdot T} = \frac{g}{k}[/tex]
[tex]e^{-k\cdot T} = \frac{\frac{g}{k} }{v_{o}+\frac{g}{k} }[/tex]
[tex]- k\cdot T = \ln \frac{\frac{g}{k} }{v_{o}+\frac{g}{k} }[/tex]
[tex]T = -\frac{1}{k}\cdot \ln \frac{\frac{g}{k} }{v_{o}+\frac{g}{k} }[/tex]
By applying algebraic handling, exponential and logarithm properties, we conclude that the equation of the time of flight (T) is [tex]T = -\frac{1}{k}\cdot \ln \frac{\frac{g}{k} }{v_{o}+\frac{g}{k} }[/tex].
To learn more on exponential functions: https://brainly.com/question/11487261
#SPJ1
