Question

A hot air balloon is descending with a velocity of 2.0 m/s straight down. At a height of 6m, a champagne bottle is opened to celebrate a successful flight, expelling the cork horizontally with a velocity of 5.0 m/s. a. What is the initial velocity (magnitude and direction) of the cork, as seen by an observer on the ground? b. What are the horizontal and vertical components of the initial velocity? c. How long is the cork in the air? d. How far away from the balloon does it land?

Answer 1

Answer:

a

   $v = 5.39 \ m/s$

b

Horizontal component

     $v_x = 5.00 \ m/s$

vertical component

     $v_y = - 2.0 \ m/s$

c

     $t = 0.921 \ s$

d

$d = 4.605 \ m$

Explanation:

Generally from the question we can deduce that he initial velocity of the cork, as seen by an observer on the ground in terms of the x  unit vector is  

     $v_x = 5.00 \ m/s$ due to the fact that the cork is moving horizontally

Generally from the question we can deduce that the vertical and horizontal  components of the initial velocity is  

       $v_y = - 2.0 \ m/s$ due to the fact that the balloon is moving downward which is the negative which will also cause the cork to move vertically with the balloon speed

Generally the  initial velocity (magnitude and direction) of the cork, as seen by an observer on the ground is mathematically represented as

       $v = \sqrt{ v^2 _x + v^2 _y }$

       $v = \sqrt{ 5^2 + (-2)^2 _y }$

        $v = 5.39 \ m/s$

Generally the  initial direction of motion as seen by the same observer is mathematically represented as

    $\theta = tan^{-1}[\frac{2}{5} ]$

    $\theta = 21.80^o$

Generally the time taken by the cork in the air before landing is mathematically represented as

       $D = ut + \frac{1}{2} g t^2$

So  D =  6 \  m from the question

     g =  9.8 \  m/s^2

     u  =  $v_y$ =  2 m/s  this because we are considering the  vertical motion

So

     $6 = 2 t + \frac{1}{2} * 9.8* t^2$

       $6 = 2 t + 4.9 t^2$

Solving using quadratic formula w have that

      $t = 0.921 \ s$

Generally the distance of the cork from the balloon is mathematically represented as

     $d = v_x * t$

    $d = 5 * 0.921$

      $d = 4.605 \ m$