- Use the kinematic equation \(h(t) = h_0 + v_0t - \frac{1}{2}gt^2\) to model the height.
- Substitute the given values: \(h_0 = 1.5\), \(v_0 = 22\), and \(\frac{1}{2}g = 4.9\).
- Set \(h(t) = 0\) for the moment the javelin hits the ground: \(-4.9t^2 + 22t + 1.5 = 0\).
- Select the matching equation from the given choices: \(\boxed{-4.9 t^2 + 22t + 1.5 = 0}\).
Explanation
- Understanding the Kinematic Formula
The problem asks for the equation that models the time \(t\) it takes for a javelin, released at an initial height of \(1.5\) meters with an initial vertical velocity of \(22\) m/s, to reach the ground. The vertical motion of an object under gravity is described by the kinematic formula:
\[h(t) = h_0 + v_0t - \frac{1}{2}gt^2\]
where:
- \(h(t)\) is the vertical position (height) at time \(t\).
- \(h_0\) is the initial height.
- \(v_0\) is the initial vertical velocity.
- \(g\) is the acceleration due to gravity, approximately \(9.8 \text{ m/s}^2\).
- \(\frac{1}{2}g \approx 4.9 \text{ m/s}^2\).
- Setting up the Equation
We are given the following values:
- Initial height \(h_0 = 1.5\) meters.
- Initial vertical velocity \(v_0 = 22\) meters per second.
- Gravitational acceleration constant \(\frac{1}{2}g = 4.9\) meters per second squared.
Substituting these values into our kinematic equation, we get:
\[h(t) = 1.5 + 22t - 4.9t^2\]
The javelin hits the ground when its height above the ground is zero, i.e., \(h(t) = 0\). Thus, we set the expression equal to zero:
\[-4.9t^2 + 22t + 1.5 = 0\]
- Selecting the Correct Equation
We evaluate the given options based on the derived equation \(-4.9t^2 + 22t + 1.5 = 0\):
- \(-4.9 t^2 + 22t = 1.5\): This is equivalent to \(-4.9 t^2 + 22t - 1.5 = 0\), which is incorrect.
- \(-16 t^2 + 22t + 1.5 = 0\): This uses a different gravitational constant (likely in ft/s\(^2\)), which is incorrect.
- \(-4.9 t^2 + 22t + 1.5 = 0\): This matches our derived equation perfectly.
- \(-4.9 t^2 + 1.5t + 22 = 0\): This has the coefficients placed incorrectly.
Therefore, the third option is the correct mathematical model for this problem. The final answer is \(\boxed{-4.9 t^2 + 22t + 1.5 = 0}\).
Examples
This model is fundamental in sports science, particularly in field events like the javelin throw, shot put, or discus. Athletes and coaches use these kinematic equations to analyze the release angle and initial velocity required to maximize the distance of a throw. By understanding how gravity affects an object's trajectory, they can calculate the optimal conditions for achieving a record-breaking performance. This type of math is also used in ballistics, architectural design for projectile safety, and even in video game physics to make movement feel natural.