A payload is released at an elevation of 500m from a hot-air balloon rising at a rate of 14 m/s

v(t)=?

How would I go about starting this problem? I know that the procedure is relatively simple but I am beyond rusty and second guess myself too much.

Any help would be greatly appreciated
gedies @ gmail (dot) com

Hi = 500

Vi = +14

v = Vi - 9.8 t
v = 14 - 9.8 t
at top, v = 0
so at top
t = 14/9.8

h = Hi + Vi t - (9.8/2) t^2
so for example max height is
h = 500+14 (14/9.8)-(9.8/2)(14^2)/9.8^2
or
h = 500 + (1/2)(14^2)/9.8 at top

To find the velocity (v(t)) of the payload released from the hot-air balloon at any time (t), you can use the basic principles of kinematics.

The key equation that relates velocity, time, and acceleration is: v(t) = v₀ + a*t, where v₀ is the initial velocity, a is the acceleration, and t is the time.

In this case, the initial velocity of the payload is the same as the velocity of the hot-air balloon, which is rising at a rate of 14 m/s. Consequently, v₀ = 14 m/s.

To determine the acceleration (a) of the payload, you need to consider the forces acting on it. At any given time, the only force acting on the payload is gravity, which causes it to accelerate downwards at a rate of 9.8 m/s².

Thus, a = -9.8 m/s² (negative because it acts in the opposite direction to the positive direction of motion).

Now that you have the values of v₀ and a, you can substitute them into the equation v(t) = v₀ + a*t.

v(t) = 14 - 9.8*t

This equation gives you the velocity of the payload at any time (t), where time starts from when the payload is released.