- The projectile motion equation is given by \(y(t) = v_{0y}t - \frac{1}{2}gt^2\).
- Using \(v_{0y} = 38\) m/s and \(g = 9.8\) m/s\(^2\), we find the expression for height is \(-4.9t^2 + 38t\).
- Setting the height equal to 0, we identify the equation \(-4.9t^2 + 38t = 0\) as the one that models when the ball hits the ground.
- The correct option is \(\boxed{-4.9 t^2 + 38 t = 0}\).
Explanation
- Analyze the Problem and Kinematic Model
The vertical motion of a projectile near the Earth's surface can be modeled using the kinematic equation for position:
\[y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2\]
where \(y(t)\) is the vertical position at time \(t\), \(y_0\) is the initial vertical position, \(v_{0y}\) is the initial vertical velocity, and \(g\) is the acceleration due to gravity. We are given that the initial velocity is \(v_{0y} = 38\) m/s and the ball starts from the ground, so the initial position \(y_0 = 0\). Using the standard value for gravity, \(g \approx 9.8\) m/s\(^2\), we find \(\frac{1}{2} g = 4.9\) m/s\(^2\).
- Formulate the Equation
To find the time it takes for the football to hit the ground, we look for the time \(t\) when the vertical position \(y(t)\) returns to \(0\). Substituting our known parameters into the kinematic equation, we obtain:
\[y(t) = 0 + 38t - 4.9t^2 = -4.9t^2 + 38t\]
Setting the vertical position to \(0\) to represent the ball hitting the ground gives us the quadratic equation:
\[-4.9t^2 + 38t = 0\]
This equation models the time \(t\) at which the football hits the ground. Comparing this with the provided options, we see that it matches the third option.
- Conclusion
The equation that correctly represents the vertical motion of the football and allows us to solve for the time it hits the ground is \(-4.9t^2 + 38t = 0\). The final answer is \(\boxed{-4.9 t^2 + 38 t = 0}\).
Examples
In physics, projectile motion is essential for understanding how objects move in space, such as a ball thrown in a sports game, a rocket launch, or even a water fountain. By modeling these movements with quadratic equations like the one used here, we can predict exactly where and when an object will land. This fundamental math concept is the foundation for engineering, aerospace science, and sports analytics, helping us design everything from safer vehicles to improved athletic equipment.