- Identify the kinematic equation \(h(t) = h_0 + v_0 t + \frac{1}{2} a t^2\).
- Substitute the given parameters: \(h_0 = 0\), \(v_0 = 30\), and \(a = -32\).
- Simplify the equation for the position to \(h(t) = -16t^2 + 30t\).
- Determine the condition for hitting the ground (\(h(t) = 0\)) to arrive at the equation \(-16t^2 + 30t = 0\), which matches the first option.
- Final answer: \(\boxed{-16t^2 + 30t = 0}\)
Explanation
- Analyzing the Motion of the Object
We analyze the motion of a ball thrown vertically upward. The standard kinematic equation for the vertical position \(h(t)\) at time \(t\) is given by:
\[h(t) = h_0 + v_0 t + \frac{1}{2} a t^2\]
Here, \(h_0\) is the initial height, \(v_0\) is the initial velocity, and \(a\) is the acceleration due to gravity. The problem states that the ball is thrown from ground level, so \(h_0 = 0\). The initial velocity is upward at \(v_0 = 30\) feet per second. Gravity acts in the downward direction, so the acceleration is \(a = -32\) feet per second squared (the standard gravitational constant in Imperial units).
- Formulating the Equation
Substituting the known values into the kinematic equation:
\[h(t) = 0 + 30t + \frac{1}{2}(-32)t^2\]
\[h(t) = 30t - 16t^2\]
The ball remains in the air until it returns to the ground, which corresponds to the height being zero, \(h(t) = 0\). Setting the expression to zero, we get:
\[-16t^2 + 30t = 0\]
This equation models the time \(t\) for which the ball is in the air.
- Identifying the Correct Equation
Comparing our derived equation \(-16t^2 + 30t = 0\) with the given options:
- \(-16t^2 + 30t = 0\)
- \(-4.9t^2 - 30t = 0\)
- \(-16t^2 - 30t = 0\)
- \(16t^2 + 30t = 0\)
The first option is the correct mathematical model for the scenario described. Thus, the final answer is the first option.
Examples
This type of kinematic equation is fundamental in engineering and sports science. For instance, coaches use these models to analyze the trajectory of a basketball shot or a volleyball serve to optimize the player's performance. By understanding the relationship between initial velocity and height, athletes can adjust their technique to achieve better accuracy and control. Mathematics provides the precise framework for these physical motions, helping us understand and predict the behavior of objects in flight.