The maximum height will occur, assuming the initial and final heights are the same, at 1/2 the total flight time.

You can then calculate the max height in terms of the initial speed and determine the initial speed necessary to reach 542m.

\(\displaystyle y(t) = v_{0}\sin\theta(t) - \dfrac{1}{2}g(t)^{2} \)

\(\displaystyle (t)(v_{0}\sin\theta-\dfrac{1}{2}g(t)^{2}) = 0\)

\(\displaystyle t = 0\)

\(\displaystyle t = \dfrac{2v_{0}\sin\theta}{g}\)

\(\displaystyle x(\dfrac{2v_{0}\sin\theta}{g}) = v_{0}\cos\theta( \dfrac{2v_{0}\sin\theta}{g})\)

\(\displaystyle v_{0}\cos(30)( \dfrac{2v_{0}\sin(30)}{9.8}) = 542\)

\(\displaystyle v_{0}(\dfrac{\sqrt{3}}{2})( \dfrac{2v_{0}(\dfrac{1}{2})}{9.8} = 542\)

\(\displaystyle v_{0}^{2} = (\dfrac{\sqrt{2}}{\sqrt{3}})(\dfrac{1}{.50})(9.8)(\dfrac{1}{2})(542)\)

\(\displaystyle v_{0}^{2} = 6133.30738\) - ??

\(\displaystyle v_{0} = \sqrt{6133.30738} = 78.31543513 = 78.3\) - ??

\(\displaystyle (t)(v_{0}\sin\theta-\dfrac{1}{2}g(t)^{2}) = 0\)

\(\displaystyle t = 0\)

\(\displaystyle t = \dfrac{2v_{0}\sin\theta}{g}\)

\(\displaystyle x(\dfrac{2v_{0}\sin\theta}{g}) = v_{0}\cos\theta( \dfrac{2v_{0}\sin\theta}{g})\)

\(\displaystyle v_{0}\cos(30)( \dfrac{2v_{0}\sin(30)}{9.8}) = 542\)

\(\displaystyle v_{0}(\dfrac{\sqrt{3}}{2})( \dfrac{2v_{0}(\dfrac{1}{2})}{9.8} = 542\)

\(\displaystyle v_{0}^{2} = (\dfrac{\sqrt{2}}{\sqrt{3}})(\dfrac{1}{.50})(9.8)(\dfrac{1}{2})(542)\)

\(\displaystyle v_{0}^{2} = 6133.30738\) - ??

\(\displaystyle v_{0} = \sqrt{6133.30738} = 78.31543513 = 78.3\) - ??

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So $t=\dfrac{v_0\sin(30)}{g}$Max height would involve y'(t)

\(\displaystyle y'(t) = v_{0}\sin(30) - 9.8t = 0\) - ?? two variables

Substitute that value for $t$ into the $y(t)$ equation, then solve for $v_0$.

\(\displaystyle y(t) = v_{0}\sin(30)(t) - \dfrac{1}{2}g(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - \dfrac{1}{2}(9.8)(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - 4.9(t)^{2} \)

\(\displaystyle y'(t) = v_{0}(\dfrac{1}{2}) - 9.8t \)

\(\displaystyle v_{0}\dfrac{1}{2}- 9.8t = 0 \)

\(\displaystyle v_{0}\dfrac{1}{2} = 9.8(t) \)

\(\displaystyle t = \dfrac{v_{0}(\dfrac{1}{2})}{9.8}\)

\(\displaystyle y(t) = v_{0}\sin(30)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(g)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = v_{0}\dfrac{1}{2}(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{(v_{0})^{2}}{\dfrac{2401}{25}}) \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{(v_{0})^{2}}{\dfrac{2401}{25}}) \)

\(\displaystyle \dfrac{2}{(v_{0})^{2}} = (\dfrac{(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{1}{\dfrac{2401}{25}}) \)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = (\dfrac{(\dfrac{1}{2})}{9.8})(\dfrac{1}{2}) - (9.8)(\dfrac{1}{\dfrac{2401}{25}}) (\dfrac{1}{2}) \)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = -\dfrac{5}{196} \)

\(\displaystyle (v_{0})^{2} = -\dfrac{196}{5}\)

\(\displaystyle (v_{0}) = \sqrt{-\dfrac{196}{5}}\) ??? - square roots cannot have a negative number inside

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - \dfrac{1}{2}(9.8)(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - 4.9(t)^{2} \)

\(\displaystyle y'(t) = v_{0}(\dfrac{1}{2}) - 9.8t \)

\(\displaystyle v_{0}\dfrac{1}{2}- 9.8t = 0 \)

\(\displaystyle v_{0}\dfrac{1}{2} = 9.8(t) \)

\(\displaystyle t = \dfrac{v_{0}(\dfrac{1}{2})}{9.8}\)

\(\displaystyle y(t) = v_{0}\sin(30)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(g)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = v_{0}\dfrac{1}{2}(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{(v_{0})^{2}}{\dfrac{2401}{25}}) \)

\(\displaystyle 2 = (\dfrac{(v_{0})^{2}(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{(v_{0})^{2}}{\dfrac{2401}{25}}) \)

\(\displaystyle \dfrac{2}{(v_{0})^{2}} = (\dfrac{(\dfrac{1}{2})}{9.8}) - (9.8)(\dfrac{1}{\dfrac{2401}{25}}) \)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = (\dfrac{(\dfrac{1}{2})}{9.8})(\dfrac{1}{2}) - (9.8)(\dfrac{1}{\dfrac{2401}{25}}) (\dfrac{1}{2}) \)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = -\dfrac{5}{196} \)

\(\displaystyle (v_{0})^{2} = -\dfrac{196}{5}\)

\(\displaystyle (v_{0}) = \sqrt{-\dfrac{196}{5}}\) ??? - square roots cannot have a negative number inside

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Alternate method: ?

\(\displaystyle y(t) = v_{0}\sin(30)(t) - \dfrac{1}{2}g(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - \dfrac{1}{2}(9.8)(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - 4.9(t)^{2} \)

\(\displaystyle y'(t) = v_{0}(\dfrac{1}{2}) - 9.8t \)

\(\displaystyle v_{0}\dfrac{1}{2}- 9.8t = 0 \)

\(\displaystyle v_{0}\dfrac{1}{2} = 9.8(t) \)

\(\displaystyle t = \dfrac{v_{0}(\dfrac{1}{2})}{9.8}\)

\(\displaystyle y(t) = v_{0}\sin(30)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(g)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = v_{0}\dfrac{1}{2}(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = (v_{0})^{2}\dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) =(v_{0})^{2} \dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)((\dfrac{v_{0})^{2}(\dfrac{1}{4})}{\dfrac{2401}{25}}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)((\dfrac{(\dfrac{1}{4})}{\dfrac{2401}{25}}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{5}{196} - \dfrac{5}{392}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{5}{396}\)

\(\displaystyle (v_{0})^{2} = \dfrac{396}{5}\)

\(\displaystyle v_{0} = \sqrt{396}{5} = \dfrac{6\sqrt{55}}{5} = 8.899438185\) ??

\(\displaystyle y(t) = v_{0}\sin(30)(t) - \dfrac{1}{2}g(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - \dfrac{1}{2}(9.8)(t)^{2} \)

\(\displaystyle y(t) = v_{0}(\dfrac{1}{2})(t) - 4.9(t)^{2} \)

\(\displaystyle y'(t) = v_{0}(\dfrac{1}{2}) - 9.8t \)

\(\displaystyle v_{0}\dfrac{1}{2}- 9.8t = 0 \)

\(\displaystyle v_{0}\dfrac{1}{2} = 9.8(t) \)

\(\displaystyle t = \dfrac{v_{0}(\dfrac{1}{2})}{9.8}\)

\(\displaystyle y(t) = v_{0}\sin(30)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(g)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = v_{0}\dfrac{1}{2}(\dfrac{v_{0}(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) = (v_{0})^{2}\dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)(\dfrac{v_{0}(\dfrac{1}{2})}{9.8})^{2} \)

\(\displaystyle y(t) =(v_{0})^{2} \dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)((\dfrac{v_{0})^{2}(\dfrac{1}{4})}{\dfrac{2401}{25}}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{1}{2}(\dfrac{(\dfrac{1}{2})}{9.8}) - \dfrac{1}{2}(9.8)((\dfrac{(\dfrac{1}{4})}{\dfrac{2401}{25}}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{5}{196} - \dfrac{5}{392}\)

\(\displaystyle \dfrac{1}{(v_{0})^{2}} = \dfrac{5}{396}\)

\(\displaystyle (v_{0})^{2} = \dfrac{396}{5}\)

\(\displaystyle v_{0} = \sqrt{396}{5} = \dfrac{6\sqrt{55}}{5} = 8.899438185\) ??

Last edited:

$y(t) = v_0 \sin(\theta) t - \dfrac {g t^2}{2}$

at $t_f$, the shell has hit the ground and $y(t)=0, t>0$

$v_0 \sin(\theta) t_f = \dfrac {g t_f^2}{2}$

$v_0 \sin(\theta) = \dfrac {g t_f}{2}$

$t_f = \dfrac {2 v_0 \sin(\theta)}{g}$

Now the maximum height occurs at $\dfrac {t_f}{2}= \dfrac {v_0 \sin(\theta)}{g}$

$y_{max}=y\left(\dfrac {t_f}{2}\right) = v_0 \sin(\theta) \left(\dfrac {v_0 \sin(\theta)}{g}\right) - \dfrac {g}{2} \left( \dfrac {v_0 \sin(\theta)}{g}\right)^2$

$y_{max}=\dfrac{v_0^2 \sin^2(\theta)}{g} -\dfrac{v_0^2 \sin^2(\theta)}{2g}= \dfrac{v_0^2 \sin^2(\theta)}{2g}$

Dumping values in

$\sin^2(30^\circ)=\dfrac 1 4$

$g=9.8$

$542 = \dfrac 1 4 \dfrac {v_0^2}{19.6}$

$542 \times 4 \times 19.6 = 42492.8 = v_0^2$

$v_0 = \sqrt{42492.8} \approx 206.14$

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