So confused Kinematics in one dimension

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A hot air balloon ascends at 8.80 m/s, while a pellet is fired upwards at 31.0 m/s from ground level. The problem involves finding two points where both the balloon and the pellet reach the same altitude simultaneously. The user initially calculated incorrect heights of 0.628 m and 3.90 m but later provided a quadratic equation approach to solve for the times when both objects are at equal heights. The correct method involves equating their height equations and solving for time using the quadratic formula.
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A hot air balloon is ascending straight up at a constant speed of 8.80 m/s. When the balloon is 12.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 31.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them has the same altitude at the same time. How far above ground level are these places?

I got 0.628 and 3.90 for my answers and they're wrong! I don't know what I did but I can't find what I did wrong! help please!
 
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how do you get 0.628 and 3.90 .show us
 
oh i got the answer! I used this equation and used the quadratic formula:

-4.9t^2+t(Velocity of pellet gun-velocity of balloon)-12=0
then plugged those two answers into the following as is:

0.628 into distance of balloon=12+speed(0.628)= 17.5 m
and distance of pellet=initial speed(3.9)-((9.8(3.9^2))/2)= 46.4 m
 
Inorder to know where those meet in the same hight you have to make equall

like pellet = gun
and you have to solve as you did above
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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