Problem 58 - Algebra 1

Question

POLE-VAULTING A pole-vaulter's approach velocity \(v\) (in feet per 19 second) and height reached \(h\) (in feet) are related by the following equation. Pole-vaulter model: \(v=8 \sqrt{h}\) If you are a pole-vaulter and reach a height of 20 feet and your opponent reaches a height of 16 feet, approximately how much faster were you running than your opponent? Round your answer to the nearest hundredth.

Step-by-Step Solution

Answer

The first pole-vaulter was running approximately 3.55 feet per second faster than the other pole-vaulter.

Step 1: Determine The Speed of The First Pole-vaulter

Substitute the height of the first pole-vaulter, 20 feet, into the pole-vaulter model \(v=8 \sqrt{h}\), which gives \(v_1=8 \sqrt{20}\).

Step 2: Determine The Speed of The Second Pole-vaulter

Similarly, substitute the height of the second pole-vaulter, 16 feet, into the model, which gives \(v_2=8 \sqrt{16}\).

Step 3: Calculate The Difference in Their Speeds

The difference in their speeds is given by the expression \(v_1 - v_2\) which means \(8 \sqrt{20} - 8 \sqrt{16} \), then round the result to the nearest hundredth to get the final answer.

Key Concepts

Radical Expressions

In mathematics, radical expressions involve roots, such as square roots, cube roots, and so on. These expressions stand for the number that, when raised to a specific power, yields the radicand—the number under the root symbol. For example, in the expression \(\sqrt{9}\), the radical sign (or square root sign) \(\sqrt{ }\) signifies we are looking for a number, which when squared (\(x^2\) gives 9. Simplifying a radical expression often involves finding a number that can be squared to result in the radicand. When we encounter a radical expression like \(8 \sqrt{h}\) in the context of pole-vaulting, the 8 is called the coefficient and amplifies the effect of the square root of height \(h\).

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For the equation \(v=8 \sqrt{h}\), v represents the velocity, and \(\sqrt{h}\) represents the square root of the height h. Square roots are a particular type of radical expression, specifically the root of index 2. For instance, \(\sqrt{16}\) equals 4 because 4 squared is 16. It's important to note that only non-negative numbers have real number square roots, as the square root of a negative number involves imaginary numbers, which are beyond the scope of most physical sports models like pole-vaulting.

Velocity and Height Relationship in Pole-Vaulting

In the realm of physics, particularly when analyzing sports like pole-vaulting, we often encounter relationships between different physical quantities. The velocity and height relationship is expressed by the equation \(v=8 \sqrt{h}\), which shows that the approach velocity \(v\) of the pole-vaulter is proportional to the square root of the height \(h\) they achieve. This implies that as a pole-vaulter increases their height, their velocity grows, but at a diminishing rate because of the square root. This relationship is vital for athletes and coaches as it helps them understand how changes in velocity impact the ultimate height achieved during the vault.

Comparing Velocities

When we are comparing velocities, we are often interested in understanding how fast one object moves relative to another. In our exercise, we compared the velocities of two pole-vaulters. By substituting their respective heights into the velocity equation, we can calculate their approach velocities. Then, by subtracting one vaulter's velocity from the other's, we found the difference in speed. This information can be extremely useful for competitive analysis or for improving an athlete's performance. Analysts and coaches use these comparisons to gauge the relative strengths and improvements needed by the athletes in their acceleration and approach technique.