Problem 44 - General Chemistry

Question

A bullet weighing 245 grains is moving at a speed of \(2.52 \times 10^{3} \mathrm{ft} / \mathrm{s} .\) Calculate the kinetic energy of the bullet in joules and in calories. One grain equals \(0.0648 \mathrm{~g}\).

Step-by-Step Solution

Answer

The kinetic energy of the bullet is approximately 4677.07 joules or 1117.84 calories.

Step 1: Convert Weight from Grains to Kilograms

First, convert the bullet's weight from grains to grams using the conversion factor. The bullet weighs 245 grains, and since one grain equals 0.0648 grams, the weight in grams is: \( 245 \times 0.0648 = 15.876 \) grams. Convert grams to kilograms by dividing by 1000: \( 15.876 \div 1000 = 0.015876 \) kg.

Step 2: Calculate Kinetic Energy in Joules

Kinetic energy (KE) is calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass in kilograms and \( v \) is the velocity in meters per second. First, convert the velocity from feet per second to meters per second: \( 2.52 \times 10^3 \times 0.3048 = 768.192 \) m/s. Now, apply the KE formula: \( KE = \frac{1}{2} \times 0.015876 \times (768.192)^2 \approx 4677.07 \) Joules.

Step 3: Convert Kinetic Energy from Joules to Calories

To convert joules to calories, use the conversion factor: 1 calorie = 4.184 joules. Divide the kinetic energy in joules by this factor: \( \frac{4677.07}{4.184} \approx 1117.84 \) calories.

Key Concepts

Conversion Factors

Understanding conversion factors is crucial when working with different units of measurement. In this exercise, weights and velocities need to be converted to appropriate units before calculating kinetic energy.

To begin, a bullet's weight is initially given in grains, which is a specialized unit that is less commonly used. Here, the conversion factor for grains to grams is applied: 1 grain equals 0.0648 grams.

• Multiply the bullet's weight in grains by 0.0648 to obtain its weight in grams.
• Further conversion is needed since kinetic energy calculations require the mass in kilograms.
• Divide the weight in grams by 1000 to convert to kilograms.

By applying these conversion steps, you ensure that the units are correctly handled, allowing for accurate calculations.

Calculation Steps

Breaking down the calculation steps for kinetic energy makes it easier to follow and understand.

First, you need to find the bullet's mass in kilograms, as explained in the conversion factors section. Then, follow these steps:

• Convert the velocity from feet per second to meters per second to comply with the standard formula units.
• Substitute the mass and velocity values into the kinetic energy formula: \[ KE = \frac{1}{2}mv^{2} \]where \( m \) represents mass and \( v \) represents velocity.
• Perform the multiplication inside the formula, taking care to square the velocity before multiplying it by half the mass.

This structured process ensures that calculations are methodically done, minimizing errors and enhancing comprehension.

Energy Units

Energy units vary across different measurement systems, thus requiring conversion for consistency. In physics and everyday applications, joules and calories are common.

Joules, the SI unit of energy, represent the work done when applying a force of one newton over one meter. This exercise begins with joules calculation from mass and velocity.

• After obtaining kinetic energy in joules, convert to calories for additional context.
• The conversion factor is 1 calorie = 4.184 joules.
• Divide the kinetic energy value in joules by 4.184 to convert it correctly to calories.

This conversion helps relate the energy value to everyday terms, offering a more tangible sense of the magnitude of a bullet's kinetic energy.

Velocity Conversion

Velocity conversion is essential for adopting consistent units in formulas, especially in kinetic energy calculations.

In this scenario, velocity is initially given in feet per second. To align with the kinetic energy formula, which relies on meters per second, perform the following conversion:

• Multiply the velocity value by 0.3048 (the number of meters in one foot).

This step ensures that your calculations are in terms of the standard metric units, facilitating accurate kinetic energy computation. Always ensure clarity by double-checking the accuracy of conversions to avoid measurement discrepancies.