Question

at a skills competition, a target is being lifted into the air by a cable at a constant speed. an archer standing on the ground launches an arrow toward the target. the system of equations below models the height, in feet, of the target and the arrow t seconds after it was fired. which statement most likely describes the situation modeled by this system?
\

$$\begin{cases} h = 8 + 2t \\\\ h = 4 + 32t - 16t^2 \\end{cases}$$

\bigcirc the arrow is fired with an initial upward velocity of 2 ft/s.
\bigcirc the arrow is fired with an initial upward velocity of 4 ft/s.
\bigcirc the arrow is fired with an initial upward velocity of 8 ft/s.
\bigcirc the arrow is fired with an initial upward velocity of 32 ft/s.

Explanation:

Step1: Recall projectile motion formula

The height of an object in projectile motion (near the surface of the Earth, neglecting air resistance) is given by the formula \( h = h_0 + v_0t - 16t^2 \), where \( h_0 \) is the initial height, \( v_0 \) is the initial upward velocity, and \( t \) is time in seconds.

Step2: Identify the arrow's height equation

The equation for the arrow's height is \( h = 4 + 32t - 16t^2 \). Comparing this with the projectile motion formula \( h = h_0 + v_0t - 16t^2 \), we can see that \( v_0 = 32 \) ft/s. The other equation \( h = 8 + 2t \) is for the target, which is moving at a constant speed (so it's a linear equation, not a projectile motion equation), so we focus on the arrow's equation.

Answer:

The arrow is fired with an initial upward velocity of 32 ft/s. (The option corresponding to this statement, e.g., if the options are labeled as A - D with this as one of them, it would be the option with "The arrow is fired with an initial upward velocity of 32 ft/s".)