Problem 5 - Applied Physics

Question

A truck with mass \(95 \overline{0}\) slugs is driven \(55.0 \mathrm{mi} / \mathrm{h}\). (a) What is its velocity in \(\mathrm{ft} / \mathrm{s}\) ? (b) What is its kinetic energy?

Step-by-Step Solution

Answer

Velocity: 80.67 ft/s; Kinetic Energy: approximately 3.09x10^8 ft·lb.

Step 1: Convert miles per hour to feet per second

To convert the speed from miles per hour to feet per second, first note there are 5280 feet in a mile and 3600 seconds in an hour. Use the conversion factor: \(1 \, \text{mile/hour} = \frac{5280}{3600} \, \text{feet/second}\). Now multiply the velocity in miles per hour by this factor: \[55.0 \, \text{mi/h} \times \frac{5280}{3600} = 80.67 \, \text{ft/s}\].

Step 2: Calculate the Kinetic Energy

The kinetic energy \( KE \) of an object is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. Given the mass \( m = 95,000 \) slugs and the velocity \( v = 80.67 \, \text{ft/s} \) from the previous step, substitute these values into the equation: \[ KE = \frac{1}{2} \times 9500 \times (80.67)^2 \]. This yields: \[ KE = \frac{1}{2} \times 95000 \times 6507.85 \approx 3.09 \times 10^8 \, \text{ft} \cdot \text{lb} \].

Key Concepts

Velocity Conversion

Converting a unit of measurement, such as velocity, allows us to better understand and compare different speeds. For instance, converting miles per hour to feet per second is common in physics to make equations more manageable. Start by recalling the conversion factors: there are 5280 feet in a mile and 3600 seconds in an hour. This means 1 mile/hour equals \( \frac{5280}{3600} \, \text{feet/second} \), which simplifies to approximately \( 1.467 \, \text{feet/second} \). To convert a speed in miles per hour to feet per second, multiply by this factor. For example, the truck's velocity given as 55 miles/hour multiplied by the conversion factor gives:

• \( 55 \times 1.467 = 80.67 \, \text{feet/second} \).

This conversion is essential when using the velocity in different physical contexts.

Mass in Slugs

Understanding mass in the context of physics often involves units like slugs, especially in the imperial system. A slug is a unit of mass used primarily in the United States. It allows for simplicity when working within the foot-pound-second system. To relate it to more familiar terms:

• 1 slug is approximately equal to 32.17 pounds-mass.
• It derives from the relationship between weight and mass (using gravitational acceleration).

Using slugs in calculations is important when dealing with kinetic energy and other dynamic physical properties.

Feet per Second

Feet per second is a unit of speed derived from the more common mile per hour. This unit becomes particularly useful in physics calculations because it aligns with other typical units, like feet for length and seconds for time. Here's why it's important:

• It facilitates easier computation in physics formulas.
• It is the standard for many physics problems, especially those dealing with motion.

When dealing with questions in physics, converting speeds to feet per second can prevent complicated conversions in the midst of solving equations.

Physics Problem Solving

Physics problem solving often involves a systematic approach to dissect the problem into solvable parts. In this kinetic energy calculation, understanding each component—like mass, velocity, and their respective units—is vital. Steps typically include:

• Identifying the known values and their units.
• Converting units if necessary to simplify the mathematical process.
• Using relevant formulas, such as kinetic energy \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.

Breaking down the problems and systematically solving each step leads to better understanding and ensures accuracy in calculations.Remember, practice and familiarity with these types of problems improve problem-solving skills.