- Identify the kinematic equation \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\).
- Substitute the given parameters: \(g = 32\), \(v_0 = 30\), and \(h_0 = 0\).
- Set the resulting height equation \(h(t) = -16t^2 + 30t\) to zero to find when the ball hits the ground.
- Match the resulting equation \(-16t^2 + 30t = 0\) with the provided options.
- The final answer is \(\boxed{-16t^2 + 30t = 0}\).
Explanation
- Analyzing the Kinematic Equation for Projectile Motion
The motion of a projectile near Earth's surface can be described by the kinematic equation for vertical position as a function of time, \(t\). The general formula for height \(h(t)\) is given by:
\[h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\]
where:
- \(g\) is the acceleration due to gravity.
- \(v_0\) is the initial vertical velocity.
- \(h_0\) is the initial height.
For this problem, we are working with units in feet and seconds. The acceleration due to gravity \(g\) is approximately \(32 \text{ ft/s}^2\). The ball is thrown upward with an initial velocity \(v_0 = 30 \text{ ft/s}\), and it starts from a platform at ground level, so the initial height \(h_0 = 0 \text{ ft}\).
- Formulating and Selecting the Correct Equation
Substituting the known values into the general formula:
- \(\frac{1}{2}g = \frac{1}{2} \times 32 = 16\).
- \(v_0 = 30\).
- \(h_0 = 0\).
The position equation becomes:
\[h(t) = -16t^2 + 30t + 0 = -16t^2 + 30t\]
The ball is in the air until it hits the ground. At the moment it hits the ground, the height is zero, so we set the position equation equal to zero:
\[-16t^2 + 30t = 0\]
This is the equation that models when the ball returns to the ground. Comparing this to the given choices, it matches the first option.
Examples
Understanding projectile motion is crucial in fields like sports science and engineering. For example, athletes like basketball players or long jumpers use these principles to maximize their performance by adjusting their launch angle and speed. Similarly, civil engineers use these equations to design safety features in sports arenas or to calculate the trajectory of objects in various industrial applications, ensuring accurate and safe outcomes.