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A quadratic equation is a mathematical expression of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) is an unknown variable. In the context of projectile motion, like our rock example, the quadratic equation describes the path of the projectile.

The parabola shape is characteristic of quadratic equations, which is evident in the graph of the height of the rock over time. Here, the equation \( S(t) = 30t - 16t^2 \) fits this form with \( a = -16 \), \( b = 30 \), and \( c = 0 \). This formula helps in predicting the height of a projectile at any given moment after being launched.

Quadratic equations serve crucial roles in modeling scenarios involving curvilinear motion, providing essential insights such as maximum height and time of flight.

Projectile motion refers to the curved trajectory that objects follow when propelled into the air under the influence of gravity, described by a quadratic equation.

In our exercise, the rock's motion is determined by the equation \( S(t) = 30t - 16t^2 \). This accounts for two components of motion: a vertical motion that propels the rock upward with an initial velocity of 30 feet per second, and the constant force of gravity pulling it downwards at 16 feet per second squared.

Key characteristics of projectile motion:

• Initial launch speed affects how high and far the rock travels.
• Gravity continuously influences the rock, creating the parabolic motion.
• The point where the rock returns to ground level marks the end of its flight.

Understanding these principles is crucial for predicting and analyzing the path the rock takes.

The vertex of a parabola represents the point at which a projectile reaches its maximum height during its motion.

To find the vertex in our equation \( S(t) = 30t - 16t^2 \), use the formula:

• \( t = -\frac{b}{2a} \)
• Substituting \( a = -16 \) and \( b = 30 \) gives \( t = \frac{30}{32} \).

This simplified computation shows that the rock achieves its highest point at approximately 0.9375 seconds.

Once the time to reach the vertex is found, substituting \( t \) back into the original equation calculates the exact height. The vertex is a critical point that defines the top of the parabola, indicating where the projectile begins its descent back to the ground.

The velocity of a projectile, like the rock, can be derived from the derivative of its height function with respect to time.

In our example, the height function is \( S(t) = 30t - 16t^2 \). The velocity function \( v(t) \) is the derivative, \( v(t) = \frac{d}{dt}(30t - 16t^2) = 30 - 32t \). This linear function represents the instantaneous rate of change of the rock's height, or its velocity, at any given moment.

Analyzing the velocity function:

• Initial velocity at \( t=0 \) is 30 feet per second, the speed at which the rock is thrown.
• As \( t \) increases, velocity decreases due to gravity (32 feet per second squared).
• When velocity reaches zero, the rock is momentarily stationary at its maximum height.
• Post-vertex, the velocity becomes negative, indicating the rock's downward motion.

Understanding the velocity function is essential for interpreting how quickly and in what direction the rock is moving throughout its flight.