Problem 123 - General Chemistry

Question

How fast (in meters per second) must an iron ball with a mass of \(56.6 \mathrm{~g}\) be traveling in order to have a kinetic energy of \(15.75 \mathrm{~J} ?\) The density of iron is \(7.87 \mathrm{~g} / \mathrm{cm}^{3} .\)

Step-by-Step Solution

Answer

The velocity is approximately 23.58 meters per second.

Step 1: Understand the Kinetic Energy Formula

The formula for kinetic energy is given by \( KE = \frac{1}{2}mv^2 \), where \( KE \) is the kinetic energy, \( m \) is mass in kilograms, and \( v \) is velocity in meters per second. Our goal is to solve for velocity \( v \).

Step 2: Convert Mass to Kilograms

The mass is given as \(56.6\mathrm{~g}\). To convert it to kilograms, divide by 1000. Therefore, \( m = \frac{56.6}{1000} = 0.0566 \mathrm{~kg}\).

Step 3: Re-arrange the Kinetic Energy Formula

To solve for \( v \), re-arrange the formula \( KE = \frac{1}{2}mv^2 \) to \( v^2 = \frac{2 \cdot KE}{m} \).

Step 4: Plug in the Known Values

Substitute \( KE = 15.75 \mathrm{~J} \) and \( m = 0.0566 \mathrm{~kg} \) into the equation \( v^2 = \frac{2 \cdot 15.75}{0.0566} \), giving \( v^2 = \frac{31.5}{0.0566} \).

Step 5: Calculate \( v^2 \) and \( v \)

Calculate \( v^2 = 556.01 \), and then take the square root to find \( v \). Thus, \( v \approx 23.58 \mathrm{~m/s} \).

Key Concepts

Mass Conversion

When dealing with physics problems, it's crucial to use consistent units. In the original exercise, the mass of the iron ball is provided as 56.6 grams. However, in physics, the standard unit of mass is kilograms.
To convert grams to kilograms, which is the required unit for the kinetic energy formula, divide the mass in grams by 1000.

• The given mass is 56.6 grams.
• To convert to kilograms, calculate: \( m = \frac{56.6}{1000} = 0.0566 \) kg.

This conversion is essential as the kinetic energy formula requires mass in kilograms. Always remember to check your units before solving the problem!

Velocity Calculation

The task is to find the velocity of the iron ball given its mass and kinetic energy. We begin with the kinetic energy formula: \( KE = \frac{1}{2} mv^2 \). Here, our objective is to isolate and solve for velocity \( v \).First, rearrange the equation to make \( v \) the subject:

• We know: \( KE = \frac{1}{2} mv^2 \)
• Re-arranging gives: \( v^2 = \frac{2 \cdot KE}{m} \)

Once rearranged, substitute the known values into this equation. For the given problem:

• \( KE = 15.75 \) Joules
• \( m = 0.0566 \) kg
• Therefore, \( v^2 = \frac{31.5}{0.0566} \)

Calculate \( v^2 \) and take the square root to find the velocity \( v \). The final step gives us approximately 23.58 \( \mathrm{m/s} \). This way, you can determine the speed required for a given kinetic energy.

Problem Solving Steps

When tackling physics problems, breaking things into clear, manageable steps is key. This helps avoid mistakes and ensures each part of the problem is properly addressed.Here's how the problem was approached in the solution:

• Step 1: Understand the given formula - In this case, \( KE = \frac{1}{2} mv^2 \). Recognize what you need to find and what you have.
• Step 2: Convert necessary values to suitable units, such as converting grams to kilograms.
• Step 3: Re-arrange the formula to isolate the variable you are solving for, namely velocity \( v \).
• Step 4: Substitute all known values into the equation.
• Step 5: Perform calculations to solve for the unknown and check your work for accuracy.

By following such a structured approach, solving physics problems becomes easier and more comprehensible. Each step logically leads to the next, creating a clear path to the solution.