A balloon is ascending at a rate of 16 ft/s at a height of 32 ft above the ground when a package is dropped.The time taken,in the absence of air resistance,for the package to reach the ground is
A)1.0 s
B)1.5 s
C)2.0 s
D)2.5 s
E)3.0 s

Question

Quizplus · Accepted Answer

We can use kinematic equations to solve the problem. Let's use the equation:
h = vi(t) + (1/2)at^2
where h is the height, vi is the initial velocity, a is acceleration (in this case, due to gravity, which is -32 ft/s^2), and t is time.

When the package is dropped, its initial velocity is 16 ft/s (the same as the balloon's rate of ascent), but its direction is downwards. Therefore, vi = -16 ft/s.

Substituting the values, we get:
0 = -16t + (1/2)(-32)t^2 + 32
Simplifying,
-16t + 16t^2 - 32 = 0
Solving for t using the quadratic formula,
t = (16 ± sqrt(16^2 - 4(16)(-32)))/(2(16))
t = (16 ± 8)/32
t = 0.5 s or t = 2 s

Since we're looking for the time taken for the package to reach the ground, we discard the 0.5 s solution, and the correct answer is 2.0 s (choice C).

A Balloon Is Ascending at a Rate of 16 Ft/s