Free Fall Problem: When Does the Parachute Open?

What This Problem Teaches

• Solving quadratic equations by taking square roots
• Interpreting domain and range in real-world contexts
• Understanding free fall motion as a quadratic function
• Connecting physics models to algebraic problem-solving
• Using function notation to model time-dependent motion

Visualizing the Problem

Solution: Method 1 — Direct Substitution and Square Root

Step 1 — Set up the equation for part (a)

We want to find when the height reaches 1000 ft, so we set h = 1000 in our function:

Step 2 — Isolate the quadratic term

Subtract 1700 from both sides:

Step 3 — Solve for t²

Divide both sides by -16:

Step 4 — Take the square root

Since we want time (which must be positive), we take the positive square root:

Step 5 — Solve part (b) for 930 ft

Following the same process for h = 930:

Step 6 — Determine domain and range

For the domain, we need when the jumper hits the ground (h = 0):

So the domain is [0, 10.31] seconds. The range is [0, 1700] feet (from ground to initial height).

Solution: Method 2 — Factoring Out the Gravitational Constant

Step 1 — Recognize the physics structure

The function h = -16t² + 1700 represents free fall where 16 = ½g (half of gravity's acceleration).

Step 2 — Factor out -16

Rewrite the equation as:

This shows that the jumper hits the ground when t² = 106.25, or t = √106.25 ≈ 10.31 seconds.

Step 3 — Use the factored form for both parts

For any target height h, we can solve:

Step 4 — Apply the formula

For h = 1000: t = √(106.25 - 62.5) = √43.75 ≈ 6.61 seconds

For h = 930: t = √(106.25 - 58.125) = √48.125 ≈ 6.94 seconds

Verification

Let's check our answers by substituting back into the original function:

For part (a):h = -16(6.61)² + 1700 = -16(43.69) + 1700 = -699 + 1700 = 1001 ✓

For part (b):h = -16(6.94)² + 1700 = -16(48.16) + 1700 = -771 + 1700 = 929 ✓

For the domain endpoint:h = -16(10.31)² + 1700 = -16(106.3) + 1700 = -1701 + 1700 = -1 ≈ 0 ✓

The small discrepancies are due to rounding in our calculations—the mathematics checks out perfectly.

Common Pitfalls

The General Pattern

This problem follows the standard free fall model:

Where h₀ is the initial height and 16 represents half of Earth's gravitational acceleration (32 ft/s²).

For any target height h_target, the time to reach that height is:

The total fall time (domain endpoint) occurs when h_target = 0:

This formula works for any object dropped from rest—whether it's a smoke jumper, a ball, or a penny dropped from a building. The key insight is that air resistance is neglected, making this a pure quadratic relationship.

Where This Shows Up in Real Life

• Skydiving and BASE jumping: Calculating deployment altitudes and fall times for safety planning
• Engineering testing: Drop tests for product durability require precise timing and impact velocity calculations
• Emergency response: Firefighting aircraft and medical helicopter drops use these calculations for accurate payload delivery