For the given problem, the height $h$ of the falling object at time $t$ is determined by the formula:

$h(t) = h_0 + v_0 t - \frac{1}{2} g t^2$

where:

• $h_0$ is the initial height (in meters),
• $v_0$ is the initial velocity (in meters per second),
• $g$ is the acceleration due to gravity ($g \approx 9.8 \, \text{m/s}^2$) on Earth, and
• $t$ is the time (in seconds) after the object begins to fall.

a) What is the applied domain of this function?

The domain of the function represents all possible values of $t$, the time variable. Since the object starts falling at time $t = 0$ and continues until it hits the ground (where $h(t) = 0$), the domain will be limited by these physical constraints.

• $t \geq 0$: Time cannot be negative, as we're considering the motion after the object starts falling.
• The upper limit of $t$ occurs when $h(t) = 0$, meaning when the object reaches the ground.

So, the domain is:

$t \in [0, t_{\text{ground}}]$

where $t_{\text{ground}}$ is the time at which the object hits the ground, which can be found by solving $h(t_{\text{ground}}) = 0$.

b) Give a scenario in which $h_0 < 0$.

A scenario where $h_0 < 0$ (the initial height is below ground level) could occur in the following situation:

Imagine a mine or tunnel below the ground. An object is dropped from a point inside this tunnel, which is beneath the Earth's surface (i.e., below the ground level). In this case, $h_0$ would be negative, indicating that the object starts falling from a position lower than the ground level, such as from a depth of 5 meters underground. If the object is released from that position, the initial height would be $h_0 = -5 \, \text{m}$, which is below ground.

Example Scenario:

• A miner drops a tool from a depth of 10 meters below ground, so $h_0 = -10 \, \text{m}$.

---

Do you want further details or have any questions on this topic? Here are some related questions you might find useful:

1. How do you solve for $t_{\text{ground}}$ when the object hits the ground?
2. How does the initial velocity $v_0$ affect the trajectory of the object?
3. What happens if the object is thrown upwards (i.e., $v_0 > 0$) rather than dropped?
4. Can the height $h(t)$ ever become negative during the motion?
5. How does the value of gravitational acceleration $g$ change on different planets?

Tip: When solving quadratic equations like $h(t) = 0$, remember to consider both roots of the equation—only the positive root will be physically meaningful in this context.