A naughty student throws a water balloon straight down with a speed of 5 ft/s from a window 40 ft above the ground.
(a) When will the balloon hit the head of an innocent 6 ft tall passerby?
(b) What will be its speed when it hits?
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We'll ignore air resistance.
The height of the balloon is
where `H_0=40` ft is the initial height, `V_0=5` ft/s is the initial downward velocity and `g=32ft/s^2` is the gravity acceleration.
The (downward) speed of the balloon is
Let's find time `t_1` for which `H(t_1)=6` ft (the height of a passerby):
This is a quadratic equation for `t_1,` and `t_1` must be >0, so
`t_1 = (-5+sqrt(25+4*16*34))/32 = (-5+sqrt(2201))/32 approx 1.31 (s).`
Speed will be 5+16*1.3=25.8 (ft/s).
The answers: (a) 1.3 s (b) 25.8 ft/s.
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