- Use the kinematic equation for vertical motion \(h(t) = h_0 + v_0 t + \frac{1}{2} a t^2\).
- Substitute the initial height \(h_0 = 0\), initial velocity \(v_0 = 38\), and acceleration \(a = -9.8\).
- Set the height to \(h(t) = 0\) for the moment the ball hits the ground, resulting in the equation \(-4.9 t^2 + 38 t = 0\).
- The correct option is \(-4.9 t^2 + 38 t = 0\).
- Final Answer: \(\boxed{-4.9 t^2 + 38 t = 0}\)
Explanation
- Analyzing the Motion Model
To determine how long it takes for a football to hit the ground, we model its vertical position using the kinematic equation for motion under constant acceleration:
\[h(t) = h_0 + v_0 t + \frac{1}{2} a t^2\]
where \(h(t)\) is the vertical position at time \(t\), \(h_0\) is the initial height, \(v_0\) is the initial vertical velocity, and \(a\) is the constant acceleration. Given that Gary kicks the ball from the ground, the initial height is \(h_0 = 0\) m. The initial upward velocity is \(v_0 = 38\) m/s, and the acceleration due to gravity acts downward, so \(a = -9.8\) m/s\(^2\).
- Substituting Known Variables
Substitute the given values into the kinematic equation:
\[h(t) = 0 + 38 t + \frac{1}{2} (-9.8) t^2\]
Simplifying the term with the acceleration gives:
\[\frac{1}{2} \times (-9.8) = -4.9\]
Thus, the equation for the vertical position is:
\[h(t) = -4.9 t^2 + 38 t\]
- Formulating the Final Equation
The football hits the ground when its vertical position is zero, which means we set \(h(t) = 0\). This leads to the quadratic equation:
\[-4.9 t^2 + 38 t = 0\]
Comparing this result to the provided options, we see that it matches the first option. Therefore, the equation that models the time \(t\) until the ball hits the ground is \(-4.9 t^2 + 38 t = 0\).
Examples
In sports physics, calculating the trajectory of a ball is fundamental for players and analysts. For instance, a soccer player adjusting their angle of kick to reach a teammate or a field goal kicker determining the hang time for a play relies on these kinematic equations. By understanding how gravity and initial velocity dictate the flight time, one can predict landing points and manage game strategies effectively.