The height of a projectile, h, is a function of the time, t it is in the air. The height in feet for t seconds
is given by the function h(t).
a. What is the domain of the function?

To determine the domain of the function \( h(t) \), which represents the height of a projectile as a function of time \( t \), we need to consider the physical context of the problem.

The domain of \( h(t) \) represents all the possible values of \( t \) for which the height is defined and meaningful. Since \( t \) represents time, it must be a non-negative value, as time cannot be negative in this context. Additionally, the time \( t \) will only be valid from the moment the projectile is launched (at \( t = 0 \)) until the moment it hits the ground.

Thus, the domain of \( h(t) \) is:

\[
\text{Domain of } h(t): \, t \in [0, T]
\]

where \( T \) is the time when the projectile hits the ground. The specific value of \( T \) depends on the details of the function \( h(t) \) (like initial velocity, height, and gravity), but generally, \( T \) is the time when \( h(t) = 0 \) after being launched.

Would you like to go into further details about determining \( T \) or have any other questions?

Here are 5 related questions you might consider:
1. How do you calculate the maximum height of the projectile?
2. What is the general form of the function \( h(t) \) for a projectile?
3. How does the initial velocity affect the time \( T \) when the projectile hits the ground?
4. What factors influence the shape of the \( h(t) \) graph?
5. How do you find the time at which the projectile reaches its maximum height?

**Tip:** Understanding the domain in physical problems often involves considering the realistic constraints of the situation, such as non-negative time and conditions when the object is in motion.

The height of a projectile, h, is a function of the time, t it is in the air. The height in feet for t seconds is given by the function h(t). What is the domain of the function?

Learn how to determine the domain of a projectile motion function, h(t). This includes key mathematical concepts like projectile motion, quadratic functions, and the domain of a function. Suitable for students in Grades 9-11, this problem helps in understanding time intervals in physics applications.

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