- Identify the vertical position model as \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\).
- Substitute the given constants \(v_0 = 12\), \(h_0 = 2\), and \(\frac{1}{2}g = 4.9\) to get \(h(t) = -4.9t^2 + 12t + 2\).
- Set the height to zero (\(h(t) = 0\)) to find the impact time.
- The correct equation is \(\boxed{-4.9t^2 + 12t + 2 = 0}\).
Explanation
- Understanding the Kinematic Equation for Vertical Motion
The vertical position \(h(t)\) of an object launched near the Earth's surface under gravity is determined by the kinematic equation for vertical motion:
\[h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\]
where \(g\) represents the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\)), \(v_0\) is the initial vertical velocity, and \(h_0\) is the initial height above the ground. In this problem, we are given:
- Initial velocity \(v_0 = 12 \text{ m/s}\)
- Initial height \(h_0 = 2 \text{ m}\)
- Gravity \(g \approx 9.8 \text{ m/s}^2\), so \(\frac{1}{2}g \approx 4.9 \text{ m/s}^2\).
- Setting up the Quadratic Equation
To model the height of the ball over time \(t\), we substitute the given values into the general equation:
\[h(t) = -4.9t^2 + 12t + 2\]
The problem asks for the time when the ball hits the ground. At this moment, the height of the ball above the ground is zero, meaning \(h(t) = 0\). Thus, we set the quadratic expression equal to zero to solve for \(t\):
\[-4.9t^2 + 12t + 2 = 0\]
- Comparing and Concluding
Comparing our formulated equation \(-4.9t^2 + 12t + 2 = 0\) with the provided options:
- \(-4.9t^2 + 12t = 2\) is equivalent to \(-4.9t^2 + 12t - 2 = 0\) (Incorrect)
- \(-4.9t^2 + 12t + 2 = 0\) (Correct)
- \(4.9t^2 + 12t + 2 = 0\) (Incorrect)
- \(-16t^2 + 12t + 2 = 0\) (Incorrect, as 16 is used when gravity is in feet/s²)
The second option is the correct mathematical model for this scenario. The final answer is \(\boxed{-4.9t^2 + 12t + 2 = 0}\).
Examples
This type of quadratic modeling is essential in sports physics and engineering. For example, coaches use these equations to predict the trajectory of a ball to optimize throwing techniques, or engineers use similar models to ensure safety when designing structures where falling objects might be a concern. By understanding how initial velocity and height interact with gravity, we can accurately predict the motion of any projectile, whether it is a baseball, a stone, or a space vehicle launch.