Question 1196745
.
use the quadratic formula to determine the times when an object launched vertically 
from the surface of the earth at an initial speed of 40m/s reaches a height of 50 meters. 
Include a sketch with the solution.
~~~~~~~~~~~~~~~~~


<pre>
The formula for the height under given conditions is

    h(t) = -5t^2 + 40t.


The coefficient at t^2 is half of the gravity acceleration with the sign "minus".

The coefficient at "t" is the given initial vertical speed of 40 m/s.


To find the time, when the object will get the height of 50 m, you should solve
this equation 

    -5t^2 + 40t = 50,

or, which is equivalent,

    t^2 - 8t + 10 = 0.


Usually, when a student obtains such a problem, it is assumed, that he (or she) just knows
the technique of solving quadratic equation using the quadratic formula

    {{{t[1,2]}}} = {{{(8 +- sqrt(8^2 - 4*1*10))/2)}}} = {{{(8 +- sqrt(24))/2}}} = {{{(8 +- 2*sqrt(6))/2}}} = {{{4 +- sqrt(6)}}}.


So, there are two solutions  {{{t[1]}}} = {{{4 - sqrt(6)}}} = 1.55 seconds (rounded), which relates to ascending move,

and  {{{t[2]}}} = {{{4 + sqrt(6)}}} = 6.45 seconds (rounded), which relates to the descending move.


The plot in (y,t)-coordinates is shown below.


    {{{graph( 400, 400, -2, 10, -10, 90,  
              -5x^2 + 40x , 50                 
)}}}


        Plot  h(t) = -5t^2 + 40t (red) and  h = 50 m (green)
</pre>

Solved.


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At this site, &nbsp;there are lessons, &nbsp;where you can find many other similar solved problems

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Introductory-lesson-on-a-projectile-thrown-shot-launched-vertically-up.lesson>Introductory lesson on a projectile thrown-shot-launched vertically up</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-projectile-moving-vertically.lesson>Problem on a projectile moving vertically up and down</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-projectile-shooted-vertically-upward.lesson>Problem on an arrow shot vertically upward</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-problems-on-an-projectile-moving-vertically-up-and-down.lesson>Problem on a ball thrown vertically up from the top of a tower</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-toy-rocket-launched-vertically-up--from-the-top-of-a-platform.lesson>Problem on a toy rocket launched vertically up from a tall platform</A>


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Read them attentively and learn how to solve this type of problems once and for all.


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&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic "<U>Projectiles launched/thrown and moving vertically up and dawn</U>".



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