Problem 7 - Applied Physics

Question

A bicycle and rider together have a mass of \(7.40\) slugs. If the kinetic energy is \(742 \mathrm{ft} \mathrm{lb}\), find the velocity.

Step-by-Step Solution

Answer

The velocity is approximately 14.16 ft/s.

Step 1: Understand the Formula for Kinetic Energy

The formula for kinetic energy (KE) is given by:\[KE = \frac{1}{2} m v^2\]where \(m\) is the mass and \(v\) is the velocity. In this problem, the KE is 742 ft lb, and the mass \(m\) is 7.40 slugs.

Step 2: Rearrange the Kinetic Energy Formula to Find Velocity

To find the velocity \(v\), we need to rearrange the kinetic energy formula to solve for \(v\):\[v^2 = \frac{2 \, KE}{m}\]Now, take the square root of both sides to solve for \(v\):\[v = \sqrt{\frac{2 \, KE}{m}}\]

Step 3: Substitute Known Values into the Formula

We have the values from the problem:\( KE = 742 \, \mathrm{ft} \, \mathrm{lb} \) and \( m = 7.40 \, \mathrm{slugs} \).Substitute these values into the formula to find \(v\):\[v = \sqrt{\frac{2 \, (742)}{7.40}}\]

Step 4: Calculate the Velocity

Compute the expression inside the square root first:\[\frac{2 \, (742)}{7.40} = \frac{1484}{7.40} \approx 200.54\]Now, take the square root of 200.54 to find \(v\):\[v \approx \sqrt{200.54} \approx 14.16 \, \mathrm{ft/s}\]

Key Concepts

Calculation of Velocity

To determine velocity from kinetic energy, we start with the kinetic energy formula, which relates mass, velocity, and kinetic energy. This formula is crucial because it shows the connection between how much energy an object has due to its motion and its speed. The velocity can be calculated by rearranging the kinetic energy equation:

• The kinetic energy formula is: \[ KE = \frac{1}{2} m v^2 \]
• For velocity (\(v\)), we manipulate the formula to:\[ v = \sqrt{\frac{2 \, KE}{m}} \]

The first step is to substitute the known values into the formula. The mass given is in slugs, and the kinetic energy is provided in foot-pounds. Using these values in the rearranged formula allows us to calculate the velocity directly. The mathematical process involves substituting these values and performing arithmetic calculations to arrive at the velocity in feet per second.

Physics Problem-Solving

Solving physics problems often involves logical reasoning, utilizing given formulas, and critical thinking. In this type of problem, the aim is to use the steps of problem-solving to discover unknown quantities like velocity.

• Begin by discerning what is known - the mass and the kinetic energy in this case.
• Identify the relationships and formulas that connect these known quantities to what you want to find, such as velocity.
• Rearrange the equations step by step, focusing on isolating the wanted variable - velocity.

Substitute numerical values, then execute the necessary mathematical operations. Make sure units are consistent throughout the process to prevent mistakes. This organized approach helps ensure accuracy and fosters understanding of the underlying physics principles.

Conservation of Energy

The principle of conservation of energy is central to solving problems involving kinetic energy. By recognizing that energy is conserved, we understand that the energy within a system remains constant unless acted on by an external force. This principle allows us to relate kinetic energy to velocity because it emphasizes that:

• The total energy is the sum of its potential and kinetic energy.
• Changes in one type of energy (like potential energy) directly affect another (like kinetic energy).

In our exercise, energy conservation implies that the kinetic energy provided can fully account for the speeding motion of the bicycle and rider. With no other forces mentioned, such as friction, all energy is kinetic, allowing us to use the energy directly in calculations.

Applied Physics

Applied physics uses physics principles to solve real-world problems, just like in this kinetic energy calculation. Understanding how an object's mass and velocity interplay to produce kinetic energy has practical applications, for example in designing safer vehicles or more efficient machinery. In this context:

• We leverage theoretical physics—like formulas and conservation laws—to tackle practical challenges.
• By computing velocity from kinetic energy, we can predict how changes in speed affect an object's energy requirements and performance.

This example highlights the importance of physics in engineering and technology, providing tools that lead to innovations and improvements in various fields. Applied physics revolves around making these abstract concepts tangible and useful in everyday life.