SOLUTION: A signal flare is fired upward with initial speed 245m/s. A stationary balloonist at the height of 1960 m sees the flare pass on the way up. How long after this will the flare pa

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Question 190294: A signal flare is fired upward with initial speed 245m/s. A stationary balloonist at the height of 1960 m sees the flare pass on the way up. How long after this will the flare pass the balloonist again on the way down?
This is for my son, and yes, I'm a few years out of practice. He's tried setting it to zero with: -4.9t^2-245t+1960=0. I think h=rt-4.9t^2 is the formula was also given to him. Can you help, please? He knows the answer is 30 seconds, but can't show the work. Thanks
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Well, first off, the equation should be -4.9t%5E2%2B245t=1960 (let me know if you need help figuring out the equation)


-4.9t%5E2%2B245t=1960 Start with the given equation.


-4.9t%5E2%2B245t-1960=0 Subtract 1960 from both sides.


-49t%5E2%2B2450t-19600=0 Multiply EVERY term by 10 to make every number a whole number.


Notice we have a quadratic equation in the form of at%5E2%2Bbt%2Bc where a=-49, b=2450, and c=-19600


Let's use the quadratic formula to solve for t


t+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


Plug in a=-49, b=2450, and c=-19600


t+=+%28-2450+%2B-+sqrt%28+6002500-4%28-49%29%28-19600%29+%29%29%2F%282%28-49%29%29 Square 2450 to get 6002500.


t+=+%28-2450+%2B-+sqrt%28+6002500-3841600+%29%29%2F%282%28-49%29%29 Multiply 4%28-49%29%28-19600%29 to get 3841600


t+=+%28-2450+%2B-+sqrt%28+2160900+%29%29%2F%282%28-49%29%29 Subtract 3841600 from 6002500 to get 2160900


t+=+%28-2450+%2B-+sqrt%28+2160900+%29%29%2F%28-98%29 Multiply 2 and -49 to get -98.


t+=+%28-2450+%2B-+1470%29%2F%28-98%29 Take the square root of 2160900 to get 1470.


t+=+%28-2450+%2B+1470%29%2F%28-98%29 or t+=+%28-2450+-+1470%29%2F%28-98%29 Break up the expression.


t+=+%28-980%29%2F%28-98%29 or t+=++%28-3920%29%2F%28-98%29 Combine like terms.


t+=+10 or t+=+40 Simplify.


So the answers are t+=+10 or t+=+40


This means that at 10 seconds the flare is at the height of 1960 m. Also, the flare is 1960 m high when the time is 40 seconds.


So the difference between these two times is 40-10=30 seconds (think of t=10 as time zero for the balloonist)


So it takes 30 seconds for the flare to come back down to the height of 1960 m.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


The height function is:



Where is the initial height of the projectile. You could argue that if it were a 6' tall man firing the flare gun, then the initial height is about 2 meters, but I suspect that this little detail was ignored for this problem, so let's just set . That gives us a height function that looks like this:



What this means is that you can calculate the instantaneous height for any value of t. But that doesn't help directly because we are given the height and we need to discover the time. Hence, the question becomes: "What is the value of t when the height is 1960 meters", or:



Put the quadratic into standard form:



Solve with the quadratic equation:













So the flare fired at time 0 gets to the balloonist at time 10 seconds and then passes him on the way back down at 40 seconds, i.e. 30 seconds later.

Extra credit: How many seconds after the flare passes the balloonist does it hit the ground?

Super-double-plus extra credit: When does the flare get as high as it will ever get, and how high is that?

John