Analyzing Diver Motion Using Quadratic Functions

Q1. Divers in Acapulco, Mexico, dive headfirst at 8 feet per second from the top of a cliff 87 feet above the Pacific Ocean. During which time period will a diver’s height exceed that of the cliff?

Background

Topic: Quadratic Functions and Projectile Motion

This question tests your understanding of how to model vertical motion using quadratic equations, specifically the height of an object over time when thrown or dropped from a certain height with an initial velocity.

Key Terms and Formulas

• Initial height ($h_0$): The starting height of the diver, which is 87 feet.

• Initial velocity ($v_0$): The diver's initial downward velocity, 8 feet per second.

• Gravity ($g$): The acceleration due to gravity, approximately $32$ feet per second squared (downward).

• Height function: $h(t) = h_0 + v_0 t - \frac{1}{2} g t^2$

Step-by-Step Guidance

1. Write the height function for the diver: $h(t) = 87 + (-8)t - \frac{1}{2} (32) t^2$

2. Simplify the equation: $h(t) = 87 - 8t - 16t^2$

3. To find when the diver's height exceeds the cliff's height, set up the inequality: $h(t) > 87$

4. Substitute the height function into the inequality: $87 - 8t - 16t^2 > 87$

5. Simplify the inequality to solve for $t$ (time): $-8t - 16t^2 > 0$

Try solving on your own before revealing the answer!