A quarterback throws a football toward a receiver from a height of 6 ft. The initial vertical velocity of the ball is $14 ft/s$. At the same time that the ball is thrown, the receiver raises his hands to a height of 8 ft and jumps up with an initial vertical velocity of $10 ft/s$.
Projectile motion formula:
[tex]h=-16 t^2+v t+h_0[/tex]
[tex]t=[/tex] time, in seconds, since the ball was thrown
[tex]h=[/tex] height, in feet, above the ground
Complete the system that models the heights of the ball and the receiver's hands over time.
[tex]\begin{array}{l}
h=-16 t^2+14 t+\square \\
h=-16 t^2+\square t+\square\\
\end{array}[/tex]
Answer (2)
The height of the football is modeled by h = − 16 t 2 + 14 t + 6 .
The height of the receiver's hands is modeled by h = − 16 t 2 + 10 t + 8 .
The completed system of equations is: h = − 16 t 2 + 14 t + 6 h = − 16 t 2 + 10 t + 8
The missing values are 6 , 10 , 8 .
Explanation
Understanding the Problem We are given the projectile motion formula h = − 16 t 2 + v t + h 0 , where h is the height, t is the time, v is the initial vertical velocity, and h 0 is the initial height. We need to complete the system of equations for the height of the ball and the height of the receiver's hands.
Finding the Equation for the Football For the football, the initial vertical velocity is v = 14 f t / s and the initial height is h 0 = 6 f t . Substituting these values into the projectile motion formula, we get h = − 16 t 2 + 14 t + 6 .
Finding the Equation for the Receiver's Hands For the receiver's hands, the initial vertical velocity is v = 10 f t / s and the initial height is h 0 = 8 f t . Substituting these values into the projectile motion formula, we get h = − 16 t 2 + 10 t + 8 .
The Complete System of Equations Therefore, the completed system of equations is:
h = − 16 t 2 + 14 t + 6
h = − 16 t 2 + 10 t + 8
Final Answer The completed system that models the heights of the ball and the receiver's hands over time is:
h = − 16 t 2 + 14 t + 6
h = − 16 t 2 + 10 t + 8
So the missing values are 6, 10, and 8.
Examples
Understanding projectile motion is crucial in sports like basketball or baseball, where players need to predict the trajectory of a ball. For example, a basketball player calculates the initial velocity and angle required to make a shot, considering the height of the hoop and the distance. Similarly, in baseball, understanding the trajectory helps fielders catch the ball and batters hit it effectively. By mastering these concepts, athletes can optimize their performance and improve their accuracy in various sports.
The heights of the ball and the receiver's hands can be modeled with the equations h = − 16 t 2 + 14 t + 6 for the football and h = − 16 t 2 + 10 t + 8 for the receiver. The missing values are 6, 10, and 8. This complete set of equations helps understand the projectile motion involved in catching a pass.
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