Question

A hot air ballo0n is ascending straight up at a constant

Answer 1

Answer: The two places altitudes are: 16.17 m and 40.67 m

Explanation:

Hi!

Lets call z to the vertical direction (z= is ground) . Then the positions of the balloon and the pellet, using the values of the velocities we are given, are:

$z_b =\text{balloon position}\\z_p=\text{pellet position}\\z_b=(7\frac{m}{s})t\\z_p=30\frac{m}{s}(t-t_0)-\frac{g}{2}(t-t_0)^2\\g=9.8\frac{m}{s^2}$

How do we know the value of t₀? This is the time when the pellet is fired. At this time the pellet position is zero: its initial position. To calculate it we know that the pellet is fired when the ballon is in z = 12m. Then:

$t_0=\frac{12}{7}s$

We need to know the when the z values of balloon and pellet is the same:

$z_b=z_p\\(7\frac{m}{s})t =30\frac{m}{s}(t-\frac{12}{7}s)-\frac{g}{2}(t-\frac{12}{7}s)^2$

We need to find the roots of the quadratic equation. They are:

$t_1=2.31s\\t_2=5.81$

To know the altitude where the to objects meet, we replace the time values:

$z_1=16,17m\\z_2=40,67m$