How to Find the Time at Which Maximum Height Is Reached Using the Vertex Formula

When analyzing the motion of an object under gravity, such as a ball thrown into the air, one key question is: At what time does the object reach its maximum height? For quadratic equations that model height over time, the answer lies in finding the vertex of the parabola represented by the equation. This allows us to determine the exact moment of peak elevation without graphing the function.

Understanding the Quadratic Model

Understanding the Context

In physics and mathematics, projectile motion is often described by a quadratic equation of the form:

$$
h(t) = at^2 + bt + c
$$

where:

  • $ h(t) $ is the height at time $ t $,
  • $ a $, $ b $, and $ c $ are constants,
  • $ a < 0 $ ensures the parabola opens downward, meaning there is a maximum point.

In our case, the height function is:

To find the time at which maximum height is reached, use the vertex formula $ t = -\frac{b}{2a} $ for the quadratic equation $ h(t) = -5t^2 + 20t + 10 $.

Key Insights

$$
h(t) = -5t^2 + 20t + 10
$$

Here, $ a = -5 $, $ b = 20 $, and $ c = 10 $. Since $ a $ is negative, the parabola opens downward, so the vertex represents the peak height and the corresponding time.

Using the Vertex Formula

To find the time $ t $ at which the maximum height is reached, use the vertex formula:

$$
t = - rac{b}{2a}
$$

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Solution: We are given 5 distinct parts: A, B, C, D, and E, and we are to count the number of permutations where A appears before B and C appears before D.

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Final Thoughts

Substituting $ a = -5 $ and $ b = 20 $:

$$
t = - rac{20}{2(-5)} = - rac{20}{-10} = 2
$$

Thus, the maximum height is achieved at $ t = 2 $ seconds.

Why This Works

The vertex formula derives from completing the square or using calculus, both confirming that the axis of symmetry of the parabola lies at $ t = - rac{b}{2a} $. This time corresponds to the peak of the motion — exactly when the upward velocity becomes zero and the object begins descending.

Real-World Application

Imagine throwing a ball straight upward. Even without graphic tools, using $ h(t) = -5t^2 + 20t + 10 $, you instantly know the ball peaks at $ t = 2 $ seconds — critical for catching it at its highest point or assessing impact timing.

Summary:
To find the time of maximum height in a quadratic motion model, apply $ t = - rac{b}{2a} $. For $ h(t) = -5t^2 + 20t + 10 $, this yields $ t = 2 $. This method simplifies vertical motion analysis and supports physics-based problem solving.