Graham and Hunter are circus performers. A cable lifts Graham into the air at a constant speed of [tex]$1.5 ft / s$[/tex]. When Graham's arms are 18 ft above the ground, Hunter, who is standing directly underneath Graham, throws Graham a ball as the cable continues to lift him higher. Hunter throws the ball from a position 5 ft above the ground with an initial velocity of [tex]$24 ft / s$[/tex]. Which system of equations can be used to model this situation?
[tex]\left\{\begin{array}{l}h=18+1.5 t \\ h=5+24 t-16 t^2\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}h=18+1.5 t \\ h=5+24 t+16 t^2\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}h=18+1.5 t \\ h=5+24 t\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}h=18+1.5 t-16 t^2 \\ h=5+24 t-16 t^2\end{array}\right.[/tex]
Answer (1)
Model Graham's height with a linear equation: h = 18 + 1.5 t .
Model the ball's height with a projectile motion equation: h = 5 + 24 t − 16 t 2 .
Combine the two equations to form a system of equations.
The correct system of equations is: { h = 18 + 1.5 t h = 5 + 24 t − 16 t 2 .
Explanation
Analyzing the Problem Let's analyze the motion of Graham and the ball separately to determine the equations that model their heights.
Modeling Graham's Height Graham is lifted at a constant speed of 1.5 f t / s starting from a height of 18 f t . Therefore, his height h at time t can be modeled by the equation: h = 18 + 1.5 t This equation represents a linear relationship between Graham's height and time.
Modeling the Ball's Height Hunter throws the ball from an initial height of 5 f t with an initial velocity of 24 f t / s . The height of the ball h at time t can be modeled using the projectile motion equation: h = h 0 + v 0 t − 2 1 g t 2 where h 0 is the initial height, v 0 is the initial velocity, and g is the acceleration due to gravity. In this case, h 0 = 5 , v 0 = 24 , and g ≈ 32 f t / s 2 . Therefore, the equation becomes: h = 5 + 24 t − 2 1 ( 32 ) t 2 = 5 + 24 t − 16 t 2
Forming the System of Equations Combining the equations for Graham's and the ball's heights, we get the following system of equations: { h = 18 + 1.5 t h = 5 + 24 t − 16 t 2
Identifying the Correct Option Comparing the derived system of equations with the given options, we find that the correct system of equations is: { h = 18 + 1.5 t h = 5 + 24 t − 16 t 2
Examples
Understanding projectile motion and linear motion is crucial in many real-world scenarios, such as designing amusement park rides. For instance, engineers must calculate the trajectory of a roller coaster car as it moves along a track, accounting for both the initial push (initial velocity) and the effect of gravity. Similarly, understanding linear motion helps in determining the constant speed of a component like a conveyor belt, ensuring items are transported efficiently and safely. By applying these physics principles, engineers can create thrilling and safe experiences for riders.
