- Identify the coefficients for the quadratic motion model: the constant term is the initial height (\(h_0 = 3\)), the coefficient of \(t\) is the initial velocity (\(v_0 = 50\)), and the coefficient of \(t^2\) is the acceleration component (-16).
- Substitute these values into the standard quadratic form \(h(t) = c_1 t^2 + c_2 t + c_3\) to form the equation.
- Conclude that the correct model is \(h(t) = -16t^2 + 50t + 3\).
- The final answer is \(\boxed{h(t) = -16t^2 + 50t + 3}\).
Explanation
- Defining the Kinematic Model
To model the height of the softball, we use the standard kinematic equation for the vertical motion of an object under constant acceleration, which is expressed as a quadratic function of time \(t\):
\[h(t) = c_1 t^2 + c_2 t + c_3\]
where:
- \(c_1\) is the coefficient related to the acceleration.
- \(c_2\) is the initial velocity of the object.
- \(c_3\) is the initial height of the object at \(t = 0\).
From the problem description, we are given:
- The initial height at time \(t = 0\) is \(h(0) = 3\) feet, so \(c_3 = 3\).
- The initial velocity is \(50\) feet per second, so \(c_2 = 50\).
- The acceleration component is given as \(-16\) ft/s\(^2\), which corresponds to the coefficient of the \(t^2\) term in the kinematic equation for vertical motion, so \(c_1 = -16\).
- Constructing the Quadratic Equation
Substituting the identified constants \(c_1 = -16\), \(c_2 = 50\), and \(c_3 = 3\) into the quadratic model \(h(t) = c_1 t^2 + c_2 t + c_3\), we obtain the equation:
\[h(t) = -16t^2 + 50t + 3\]
This equation represents the height of the softball \(h\) in feet as a function of time \(t\) in seconds. This matches the standard form for vertical projectile motion where the gravitational acceleration is accounted for in the quadratic term.
- Conclusion and Matching the Result
Comparing our derived equation \(h(t) = -16t^2 + 50t + 3\) with the given options, we identify that the correct quadratic model is:
\[h(t) = -16t^2 + 50t + 3\]
This model correctly accounts for the initial height of 3 feet, the initial upward velocity of 50 feet per second, and the downward acceleration of -16 feet per second squared. The final answer is \(\boxed{h(t) = -16t^2 + 50t + 3}\).
Examples
This kinematic model is essential for sports physics, allowing pitchers, coaches, and sports analysts to predict the trajectory of a softball. By knowing the initial height, launch velocity, and acceleration due to gravity, one can determine the time the ball reaches its peak height or when it will cross the home plate. Similar quadratic equations are used in engineering to calculate the paths of projectiles, from launching drones to designing amusement park rides, ensuring safety and precision in motion planning.