Problem 75 - Applied Calculus

Question

GENERAL: Impact Velocity If a marble is dropped from a height of \(x\) feet, it will hit the ground with velocity \(v(x)=\frac{60}{11} \sqrt{x}\) miles per hour (neglecting air resistance). Use this formula to find the velocity with which a marble will strike the ground if it is dropped from the height of the tallest building in the United States, the 1776 -foot One World Trade Center in New York City.

Step-by-Step Solution

Answer

The marble will strike the ground at approximately 229.046 mph.

Step 1: Identify Given Values

We are given the height from which the marble is dropped, which is the height of the One World Trade Center, 1776 feet. We need to find the velocity using the given formula for the height \( x = 1776 \) feet.

Step 2: Substitute Heights into Formula

Use the formula for impact velocity: \( v(x) = \frac{60}{11} \sqrt{x} \). Substitute \( x = 1776 \) into the formula: \[ v(1776) = \frac{60}{11} \sqrt{1776} \]

Step 3: Calculate the Square Root

Find the square root of 1776. \[ \sqrt{1776} \approx 42.133 \]

Step 4: Compute Velocity

Substitute the square root value back into the velocity formula: \[ v(1776) = \frac{60}{11} \times 42.133 \] Calculate the multiplication: \[ v(1776) \approx \frac{60}{11} \times 42.133 \approx 229.046 \]

Step 5: Interpret Result

The computed velocity means that the marble will strike the ground with a velocity of approximately 229.046 miles per hour.

Key Concepts

Velocity Formula

The velocity formula used to calculate the impact velocity of a falling marble is given by \( v(x) = \frac{60}{11} \sqrt{x} \). This formula determines the final speed of an object as it hits the ground, neglecting air resistance. Here, \( v(x) \) represents the velocity in miles per hour, and \( x \) is the height in feet from which the marble is dropped.

This formula incorporates the concept of gravity's effect on a falling object's speed. While the marble accelerates due to gravity, it gains velocity proportional to the square root of the height. Thus, the formula suggests that as the height increases, the impact velocity increases, but in a non-linear manner.

Understanding this formula allows you to predict how fast an object will hit the ground from any given height. It's a useful tool in physics for analyzing motion under the force of gravity.

Square Root Calculation

Calculating the square root is a crucial step in finding the impact velocity using the velocity formula. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). In the context of our exercise, we need to determine \( \sqrt{1776} \) as part of our substitution into the velocity formula.

Using a calculator, we found that \( \sqrt{1776} \approx 42.133 \). This result is then used in the velocity calculation of the marble falling from the One World Trade Center height.

Learners often use estimation or calculators to find square roots, especially for non-perfect squares. It's essential to comprehend this because just estimating can affect precision, particularly in scientific calculations where accuracy is key.

One World Trade Center Height

One World Trade Center, standing at a soaring height of 1776 feet, is the tallest building in the United States. This height is significant in our exercise as it serves as the reference point (\( x = 1776 \) feet) for calculating the impact velocity.

This particular height is historically significant as it reflects the year 1776, marking American independence. Such monumental heights offer exciting opportunities to investigate real-world physics problems involving gravity and motion.

Applying the marble's drop from such a great height allows us to see the magnitude of velocities that can be reached under gravitational pull. Understanding these concepts provides insight into both practical and academic perspectives on physics, motion, and engineering.