Your friend is an environmentalist who is living in a tree for the summer. You are helping provide her with food, and you do so by throwing small packages up to her tree house. If her tree house is 27 m above the ground, what is the minimum (initial) speed you must use when throwing packages up to her?

To find the minimum speed you must use when throwing packages up to your friend's treehouse, we can use the concept of projectile motion.

Projectile motion involves the horizontal and vertical components of an object's motion. In this case, we are interested in the vertical component, as you need to throw the packages upward to reach the height of the treehouse.

To determine the minimum speed, we need to consider the height of the treehouse and the effects of gravity. The minimum speed occurs when the package reaches the highest point of its trajectory, just before it starts descending.

We can use the following equation to find the vertical component of the velocity for the highest point of the trajectory:

v^2 = u^2 - 2gh

Where:
v is the final vertical velocity (0 m/s at the highest point),
u is the initial vertical velocity (the speed you throw the package upward),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
h is the height of the treehouse (27 m).

Since the final vertical velocity at the highest point is 0 m/s, the equation becomes:

0 = u^2 - 2gh

Simplifying the equation gives:

2gh = u^2

Now, we can substitute the values into the equation:

2 * 9.8 * 27 = u^2

u^2 = 529.2

Taking the square root of both sides, we get:

u ≈ 23.01 m/s

Therefore, the minimum (initial) speed you must use when throwing packages up to the treehouse is approximately 23.01 m/s.