Question

A helium filled balloon (mass = 0.2 kg) is rising in the air with the buoyant force = Fb = [1.29e^(-1.21h)]g where h is in km.


How would I go about finding an expression for the velocity as a function of height by integrating acceleration?

Answer 1

Answer on Question #48633, Physics, Computational Physics

A helium filled balloon (mass = 0.2 kg) is rising in the air with the buoyant force = Fb = [1.29e^(-1.21h)]g where h is in km.

How would I go about finding an expression for the velocity as a function of height by integrating acceleration?

Solution:

$$F _ {b} = [ 1. 2 9 e ^ {- 1. 2 1 h} ] g$$

The equation of motion

$$m a = F _ {b} - m g$$

Thus,

$$a = \frac {F _ {b}}{m} - g = \frac {[ 1 . 2 9 e ^ {- 1 . 2 1 h} ] g}{m} - g = g \left(\frac {1 . 2 9}{m} e ^ {- 1. 2 1 h} - 1\right)$$ $$a = g (6. 4 5 e ^ {- 1. 2 1 h} - 1)$$ $$a (h) = \frac {d v}{d t} = \frac {d v}{d h} \frac {d h}{d t} = v \frac {d v}{d h}$$

Thus, integrating

$$\int_ {0} ^ {h} a (h) d h = \int_ {0} ^ {v} v d v$$ $$g \int_ {0} ^ {h} (6. 4 5 e ^ {- 1. 2 1 h} - 1) d h = \int_ {0} ^ {v} v d v$$ $$g (6. 4 5 e ^ {- 1. 2 1 h} - 1) \big | _ {0} ^ {h} = \frac {v ^ {2}}{2}$$ $$g (- h - 5. 3 3 0 5 8 e ^ {- 1. 2 1 h} + 5. 3 3 0 5 8) = \frac {v ^ {2}}{2}$$ $$v = \sqrt {2 g (5 . 3 3 - h - 5 . 3 3 e ^ {- 1 . 2 1 h})}$$

http://www.AssignmentExpert.com/