SOLUTION: A signal flare is fired upward from ground level at an initial speed of 294 m/s. A balloon is at 2450m and passes it. How long will it be before the flare passes the balloon going

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A signal flare is fired upward from ground level at an initial speed of 294 m/s. A balloon is at 2450m and passes it. How long will it be before the flare passes the balloon going       Log On


   

Question 295516: A signal flare is fired upward from ground level at an initial speed of 294 m/s. A balloon is at 2450m and passes it. How long will it be before the flare passes the balloon going down?
(The answer is 40 sec but I don't understand how to get that)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The flare is acting under the force of gravity with an initial upward velocity.
+a=dv%2Fdt=-g+
+v=dy%2Fdt=-gt%2BC1
+y=-%281%2F2%29gt%5E2%2BC1%2At%2BC2
where g is the gravitational constant (9.8 m/s^2), y is the flare's position from the ground (y=0 on the ground and moves positively upwards), t is time. C1 and C2 are constants that you solve for using your initial conditions.
dy%2Fdt is velocity, the first derivative of position.
dv%2Fdt is acceleration, the second derivative of position, first derivative of velocity.
You know the initial velocity (t=0).
+-9.8t%2BC1=294
+-9.8%280%29%2BC1=294
+highlight%28+C1=294%29
.
.
.
You also know the initial position (t=0). The flare is on the ground (y=0).
++y=-%281%2F2%29gt%5E2%2BC1%2At%2BC2
+-%281%2F2%29%289.8%29%280%29%5E2%2B294%280%29%2BC2=0
+highlight%28+C2=0%29
Now you have the complete equation for the flare's position as a function of time.
+highlight_green%28y%28t%29=-4.9t%5E2%2B294t%29
Solve for t when the flare is at the balloon position,y=2450.
+2450=-4.9t%5E2%2B294t
+4.9t%5E2-294t%2B2450=0
Use the quadratic formula,
+t=%28294+%2B-+sqrt%28294%5E2-4%282450%29%284.9%29%29%29%2F%282%284.9%29%29
+t=%28294+%2B-+sqrt%2886436-48020%29%29%2F%289.8%29
+t=%28294+%2B-+sqrt%2838416%29%29%2F%289.8%29
+t=%28294+%2B-+196%29%2F%289.8%29
+t1=%28490%29%2F%289.8%29=10
+t2=%2898%29%2F%289.8%29=50
At time t=10 seconds, the flare passes the balloon going up and at t=50 seconds the flare passes the balloon going down. So 40 seconds after the flare passes the balloon going up, it will pass the balloon going down.