To model the motion of the arrow in this situation, we can assume that its trajectory follows a parabolic path. The general form of a parabolic equation is:

$y(t) = at^2 + bt + c$

Where:

• $y(t)$ is the height of the arrow (in feet) at time $t$ (in seconds),
• $a$, $b$, and $c$ are constants we need to determine based on the given data.

Step 1: Use the initial conditions

• At $t = 0$, the arrow starts at a height of 5.4 feet: $y(0) = 5.4$ Substituting into the equation: $5.4 = a(0)^2 + b(0) + c$ This simplifies to: $c = 5.4$ So, the equation becomes: $y(t) = at^2 + bt + 5.4$

Step 2: Use the second condition

• At $t = 0.3$, the height is 5.5 feet: $y(0.3) = 5.5$ Substituting into the equation: $5.5 = a(0.3)^2 + b(0.3) + 5.4$ Simplifying: $5.5 = a(0.09) + b(0.3) + 5.4$ Subtract 5.4 from both sides: $0.1 = 0.09a + 0.3b$ This is our second equation: $0.1 = 0.09a + 0.3b \quad \text{(Equation 1)}$

Step 3: Use the third condition

• At $t = 2$, the height is 5 feet: $y(2) = 5$ Substituting into the equation: $5 = a(2)^2 + b(2) + 5.4$ Simplifying: $5 = 4a + 2b + 5.4$ Subtract 5.4 from both sides: $-0.4 = 4a + 2b$ Divide the entire equation by 2: $-0.2 = 2a + b \quad \text{(Equation 2)}$

Step 4: Solve the system of equations

We now have the system of two equations:

1. $0.1 = 0.09a + 0.3b$
2. $-0.2 = 2a + b$

To solve, let's first solve for $b$ in Equation 2: $b = -0.2 - 2a$ Substitute this into Equation 1: $0.1 = 0.09a + 0.3(-0.2 - 2a)$ Simplifying: $0.1 = 0.09a - 0.06 - 0.6a$ $0.1 + 0.06 = 0.09a - 0.6a$ $0.16 = -0.51a$ Solving for $a$: $a = \frac{-0.16}{0.51} \approx -0.3137$

Now substitute this value of $a$ into the expression for $b$: $b = -0.2 - 2(-0.3137) \approx -0.2 + 0.6274 \approx 0.4274$

Step 5: Write the final equation

Now that we have values for $a$, $b$, and $c$, the equation of the arrow’s trajectory is:

$y(t) = -0.3137t^2 + 0.4274t + 5.4$

This is the equation that models the motion of the arrow.

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Would you like further clarification on any part of this solution, or would you like to explore more questions related to projectile motion? Here are some suggestions:

1. How would the equation change if the initial speed of the arrow were different?
2. What would happen to the height equation if air resistance were considered?
3. How can we model the arrow's horizontal motion in addition to its vertical motion?
4. What are the implications of the negative value of $a$ in this model?
5. How do we calculate the maximum height the arrow reaches using this equation?

Tip: To estimate the time it takes for the arrow to reach the ground, solve for $t$ when $y(t) = 0$.