The problem provided uses the position equation:

$s = -16t^2 + v_0 t + s_0$

where:

• $s$ is the height of the object in feet,
• $v_0$ is the initial velocity of the object in feet per second,
• $s_0$ is the initial height in feet,
• $t$ is the time in seconds.

The aircraft drops the supply package from 550 feet, and the questions are:

(a) How long does it take for the package to hit the ground?

Given the height $s_0 = 550$ feet and assuming no initial vertical velocity ($v_0 = 0$), the equation simplifies to:

$s = -16t^2 + 550$

The package hits the ground when $s = 0$:

$0 = -16t^2 + 550$

Solving for $t$:

$16t^2 = 550$ $t^2 = \frac{550}{16}$ $t = \sqrt{\frac{550}{16}}$

I'll calculate this.The time it takes for the supply package to hit the ground is approximately 5.863 seconds.

(b) How far does the package travel horizontally?

The aircraft is flying at 158 miles per hour. To find the horizontal distance traveled, we need to convert this speed to feet per second. The conversion factor is:

$1 \, \text{mile per hour} = 1.467 \, \text{feet per second}$

The horizontal distance $d$ is given by:

$d = \text{horizontal speed} \times t$

Let's calculate the horizontal distance.The supply package travels approximately 1,359.0 feet horizontally during its descent.

Would you like more details on any of the steps or calculations?

Here are 5 related questions you can explore:

1. How would the result change if the initial velocity $v_0$ was not zero?
2. How is the trajectory of the package affected by wind resistance?
3. What would happen if the aircraft were flying at a higher altitude?
4. How long would it take if the gravitational constant changed due to being on a different planet?
5. How far would the package travel horizontally if the speed of the aircraft increased by 20%?

Tip: Always check unit consistency (e.g., converting miles per hour to feet per second) to avoid calculation errors.