A quarterback throws a football toward a receiver from a height of 6 ft. The initial vertical velocity of the ball is [tex]$14 ft / s$[/tex]. 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 [tex]$10 ft / s$[/tex].
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]$\begin{array}{l}
h=-16 t^2+14 t+\square \\
h=-16 t^2+\square+\square
\end{array}$[/tex]
After [tex]$\square$[/tex] seconds, the receiver's hands and the ball will be at the same height.
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 .
Set the two equations equal to each other and solve for t .
The receiver's hands and the ball will be at the same height after 2 1 seconds.
Explanation
Problem Analysis Let's analyze the given problem. We have two objects, a football and a receiver's hands, both moving under the influence of gravity. We are given the initial height and initial vertical velocity for both. We need to complete the equations that model their heights over time and find the time when they are at the same height.
Completing the Equations The height of the football is given by the equation h = − 16 t 2 + 14 t + h 0 , where h 0 is the initial height of the football, which is 6 ft. So, the equation for the height of the football is: h = − 16 t 2 + 14 t + 6 The height of the receiver's hands is given by the equation h = − 16 t 2 + v t + h 0 , where v is the initial vertical velocity of the receiver's hands, which is 10 f t / s , and h 0 is the initial height of the receiver's hands, which is 8 ft. So, the equation for the height of the receiver's hands is: h = − 16 t 2 + 10 t + 8
Setting the Equations Equal Now we want to find the time t when the height of the football and the height of the receiver's hands are equal. We set the two equations equal to each other: − 16 t 2 + 14 t + 6 = − 16 t 2 + 10 t + 8
Simplifying the Equation We can simplify this equation by adding 16 t 2 to both sides: 14 t + 6 = 10 t + 8
Further Simplification Now, subtract 10 t from both sides: 4 t + 6 = 8
Isolating the Variable Subtract 6 from both sides: 4 t = 2
Solving for Time Finally, divide both sides by 4 to solve for t :
t = 4 2 = 2 1
Final Answer Therefore, the receiver's hands and the ball will be at the same height after 2 1 seconds.
Examples
Imagine you're timing a jump to catch a ball. Knowing the heights and velocities, you can predict when the ball and your hands will meet. This is similar to how engineers design automated systems to intercept objects in manufacturing or logistics. By modeling the motion, they can ensure precise timing for successful interactions.
The height of the football and the receiver’s hands can be modeled by the equations h = − 16 t 2 + 14 t + 6 and h = − 16 t 2 + 10 t + 8 , respectively. By setting these equations equal to find when they are the same height, we find that they meet after 2 1 seco n d s . This demonstrates how projectile motion can be analyzed in a real-world scenario.
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