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To find the value of \( a \) in the quadratic function \( h(t) = at^2 + bt + c \), representing the height of the ball over time, let’s analyze the information given:

1. **Initial Condition**: The ball is thrown from a height of \( 3 \) meters, so when \( t = 0 \), \( h(0) = 3 \).
   \[
   h(0) = a(0)^2 + b(0) + c = 3 \Rightarrow c = 3
   \]

2. **Maximum Height**: The ball reaches a maximum height of \( 5 \) meters. This maximum occurs at the vertex of the parabola. 
   - Since the ball is thrown and lands back at \( h = 3 \) after \( t = 2 \) seconds, the vertex (maximum height) occurs at the midpoint, \( t = 1 \) second.
   - We know \( h(1) = 5 \), so we can substitute:
     \[
     h(1) = a(1)^2 + b(1) + 3 = 5
     \]
     Simplifying this, we get:
     \[
     a + b + 3 = 5
     \]
     \[
     a + b = 2 \quad \text{(Equation 1)}
     \]

3. **Condition at \( t = 2 \)**: The ball returns to the height of \( 3 \) meters when \( t = 2 \).
   \[
   h(2) = a(2)^2 + b(2) + 3 = 3
   \]
   Simplifying this:
   \[
   4a + 2b + 3 = 3
   \]
   \[
   4a + 2b = 0
   \]
   Dividing by \( 2 \):
   \[
   2a + b = 0 \quad \text{(Equation 2)}
   \]

Now, we can solve the system of linear equations formed by Equation 1 and Equation 2:
\[
a + b = 2
\]
\[
2a + b = 0
\]

Subtract the first equation from the second:
\[
(2a + b) - (a + b) = 0 - 2
\]
\[
a = -2
\]

So, the value of \( a \) is \( \mathbf{-2} \).

Would you like further details on the solution steps, or do you have any questions? 

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Here are five additional questions related to this topic:

1. What is the physical interpretation of the coefficient \( a \) in this quadratic function?
2. How can we find the value of \( b \) after determining \( a \)?
3. What does the vertex of a parabola represent in real-world physics problems like this?
4. If the ball reaches a maximum height in 1 second, what is the initial velocity of the ball?
5. How would the quadratic function change if the ball were thrown from a height of \( 5 \) meters instead of \( 3 \)?

**Tip:** In a quadratic function representing vertical motion, the \( a \) term is typically negative, as it accounts for the downward acceleration due to gravity.

A ball is thrown from 3 m off the ground and reaches a maximum height of 5 m. Assume that the ball was released from the point (0, 3) in the xy-plane. The ball returns to a height of 3 m after 2 seconds. Let h(t) = at^2 + bt + c be the quadratic function which represents the height of the ball after t seconds. What is the value of a?

Solve a quadratic equation problem involving the height of a ball thrown from 3 meters, reaching a maximum of 5 meters, and returning to its original height after 2 seconds. Learn how to calculate the coefficient 'a' using quadratic functions, vertex properties, and kinematics.

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