document.write( "Question 974255: A rocket was fired with an initial velocity of 25 meters per second from a platform 10m above ground.
\n" ); document.write( "a)when will the rocket reach its maximum height?
\n" ); document.write( "b)What is the maximum height of the rocket in the air?
\n" ); document.write( "c)For how long does the rocket stay at or above 20 meters?
\n" ); document.write( "d)When does the rocket strike the ground? \n" ); document.write( "

Algebra.Com's Answer #596237 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
If you are not given the function that represents height as a function of time, this is a physics problem, rather than an algebra problem.
\n" ); document.write( "A friendly physics problem would say that you can use $\"g=10\"$ $\"m%2Fs%5E2\"$ .
\n" ); document.write( "Lacking such show of friendship, we are obligated to use the cumbersome $\"g=9.8\"$ $\"m%2Fs%5E2\"$ that is a better approximation of $\"g\"$ on/near Earth's surface.
\n" ); document.write( "
\n" ); document.write( "If we could use $\"g=10\"$ $\"m%2Fs%5E2\"$ ,
\n" ); document.write( "parts a) and b)are easy mental math:
\n" ); document.write( "a) The rocket reach its maximum height when its upwards velocity becomes $\"0\"$ ,
\n" ); document.write( "and if it starts at $\"%2225+m+%2F+s%22\"$ ,
\n" ); document.write( "decreasing at a rate of $\"%2210+m+%2F%22\"$ $\"s%5E2=%2210+m+%2F+s+%2Fs%22\"$ ,
\n" ); document.write( "that would take $\"%2225+m+%2F+s%22%2F%2210+m+%2F+s+%2Fs%22=2.5s\"$
\n" ); document.write( "b)Since the upwards velocity was decreasing linearly (at a constant rate),
\n" ); document.write( "the average upwards velocity for the upwards flight is the average of the initial and final velocities,
\n" ); document.write( " $\"%28%2225+m+%2F+s%22%2B%220+m+%2F+s%22%29%2F2=%2212.5+m+%2F+s%22\"$ .
\n" ); document.write( "during the $\"2.5s\"$ upwards flight with an average velocity of $\"%2212.5+m+%2F+s%22\"$ ,
\n" ); document.write( "the rocket traveled $\"2.5s%28%2212.5+m+%2F+s%22%29=31.25m\"$ .
\n" ); document.write( "Since the rocket started its upwards flight at a height of $\"10m\"$ ,
\n" ); document.write( "its maximum height is $\"10m%2B31.25m=41.25m\"$ .
\n" ); document.write( "
\n" ); document.write( "If you must use $\"g=9.8\"$ $\"m%2Fs%5E2\"$ ,
\n" ); document.write( "the calculations are the same,
\n" ); document.write( "but I suggest using a calculator.
\n" ); document.write( "
\n" ); document.write( "For the rest,
\n" ); document.write( " $\"h\"$ = height of the rocket above the ground, in m
\n" ); document.write( " $\"h%5B0%5D\"$ = initial height of the rocket above the ground, in m $\"t\"$ = time since launch, in seconds
\n" ); document.write( " $\"v%5B0%5D\"$ = initial velocity, in m/s
\n" ); document.write( " $\"g\"$ = acceleration of gravity, in m/s/s.
\n" ); document.write( " $\"v\"$ = upwards velocity, in m/s
\n" ); document.write( " $\"v=v%5B0%5D-gt\"$
\n" ); document.write( "The average velocity in the interval [0,t] is
\n" ); document.write( " $\"%28v%5B0%5D%2Bv%29%2F2=%28v%5B0%5D%2Bv%5B0%5D-gt%29%2F2=v%5B0%5D-gt%2F2\"$
\n" ); document.write( "and consequently, the gain in height in the interval [0,t] is
\n" ); document.write( " $\"DELTA\"$ $\"h=t%28v%5B0%5D-gt%2F2%29=v%5B0%5Dt-gt%5E2%2F2\"$ , so
\n" ); document.write( " $\"h=h%5B0%5D%2Bv%5B0%5Dt-gt%5E2%2F2\"$
\n" ); document.write( "The above equation can be used to find the answers to parts c) and d) .
\n" ); document.write( "
\n" ); document.write( "If I can use $\"g=%2210+m+%2F+s+%2F+s%22\"$ ,
\n" ); document.write( "substituting that, along with $\"h%5B0%5D=0\"$ and $\"v%5B0%5D=25\"$ , i get
\n" ); document.write( " $\"h=10%2B25t-5t%5E2\"$
\n" ); document.write( "c) $\"h=10%2B25t-5t%5E2%3E=20\"$ --> $\"10%2B25t-5t%5E2-20%3E=0\"$ --> $\"-5t%5E2%2B25t-10%3E=0\"$ --> $\"-t%5E2%2B5t-2%3E=0\"$
\n" ); document.write( "Since the solutions to $\"-t%5E2%2B5t-2=0\"$ are $\"t=%285+%2B-+sqrt%2833%29%29%2F2\"$
\n" ); document.write( " $\"-t%5E2%2B5t-2%3E=0\"$ ---> $\"%285+%2B-+sqrt%2833%29%29%2F2%3C=t%3C=%285+%2B-+sqrt%2833%29%29%2F2\"$
\n" ); document.write( "(An approximate decimal value may be expected as an answer).
\n" ); document.write( "d) $\"h=10%2B25t-5t%5E2=0\"$ --> $\"-t%5E2%2B5t%2B2%3E=0\"$
\n" ); document.write( "The two solutions are $\"t=%28-5+%2B-+sqrt%2833%29%29%2F2\"$ , but
\n" ); document.write( " $\"t=%28-5+-+sqrt%2833%29%29%2F2%3C0\"$ does not make sense,
\n" ); document.write( "so the rocket gets down to ground level at $\"t=%28-5+%2B+sqrt%2833%29%29%2F2\"$ .
\n" ); document.write( "(An approximate decimal value may be expected as an answer).
\n" ); document.write( "If you must use $\"g=9.8\"$ $\"m%2Fs%5E2\"$ ,
\n" ); document.write( "the calculations are the similar,
\n" ); document.write( "but I suggest using a calculator. \n" ); document.write( "

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