SOLUTION: A bullet is shot straight up into the air from ground level. It reaches a maximum height at h= 821m. calculate the initial speed of the bullet, in m/s. Using the previous res

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Question 1093131: A bullet is shot straight up into the air from ground level. It reaches a maximum height at h= 821m.
calculate the initial speed of the bullet, in m/s.
Using the previous results, calculate the total flight time, t, in seconds, until the bullet reaches the ground.
Answer by ikleyn(52747) About Me  (Show Source):
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Solution 1     (Physics) 

Apply the energy conservation law:

%28mv%5E2%29%2F2 = mgH,


where m is the mass of the body, v is its initial vertical speed (m/s),

g is the gravity acceleration (9.81 m/s^2), H is the maximal height.


Canceling m in both sides and simplifying, you get

v%5E2 = 2gH,   or   v =  = 126.9 m/s.


By knowing the initial speed, the physicist momentarily will find the time to reach the maximum height

t = v%2Fg = 126.9%2F9.81 = 12.9 seconds.

and the total time to get the ground is twice of this, i.e. 25.8 seconds.


Solution 2     (School math) 

Your expression for the height above the ground as a function of time is this quadratic function 

h(t) = -4.9%2At%5E2+%2B+v%2At.


The maximum is achieved at  t = -b%2F%282a%29 = -v%2F%28-2%2A4.9%29 = v%2F9.8    and is equal to

H = -4.9%2A%28v%2F9.8%29%5E2+%2B+v%2A%28v%2F9.8%29 = %28-4.9%2F9.8%29%2A%28v%5E2%2F9.8%29+%2B+%28v%5E2%2F9.8%29 = %28-1%2F2%29%2A%28v%5E2%2F9.8%29+%2B+%28v%5E2%2F9.8%29 = %281%2F2%29%2A%28v%5E2%2F9.8%29.


So, you obtain practically the same equation 

H = %281%2F2%29%2A%28v%5E2%2F9.8%29,   or   v%5E2 = 2*9.8*H,   or   v = sqrt%282%2A9.8%2AH%29 = sqrt%282%2A9.8%2A821%29 = 126.9 m/s,

and the same answer for vertical speed.

Solved.

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To wider your horizon and to get a solid base regarding problems on projectile launched vertically up,  see the lessons
    - Problem on a projectile moving vertically up and down
    - Problem on an arrow shot vertically upward
    - Problem on a ball thrown vertically up from the top of a tower
    - Problem on a toy rocket launched vertically up from a tall platform

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

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


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.