Problem 12 - College Physics

Question

\(\cdot\) It takes 4.186 \(\mathrm{J}\) of energy to raise the temperature of 1.0 \(\mathrm{g}\) of water by \(1.0^{\circ} \mathrm{C}\) (a) How fast would a 2.0 \(\mathrm{g}\) cricket have to jump to have that much kinetic energy? (b) How fast would a 4.0 \(\mathrm{g}\) cricket have to jump to have the same amount of kinetic energy?

Step-by-Step Solution

Answer

The 2.0 g cricket must jump at 64.7 m/s, while the 4.0 g cricket must jump at 45.7 m/s.

Step 1: Understanding Kinetic Energy

Kinetic energy (KE) is given by the formula: \( KE = \frac{1}{2}mv^2 \). Here, \( m \) is the mass of the object (in kg), and \( v \) is the velocity (in m/s). We need to find the velocity for which this energy equals 4.186 J.

Step 2: Converting Mass

Firstly, convert the mass of the crickets from grams to kilograms since the standard unit for mass in physics is kilograms. For the 2.0 g cricket: \[ m_1 = 2.0 \text{ g} \times \frac{1}{1000} \text{ kg/g} = 0.002 \text{ kg} \]For the 4.0 g cricket:\[ m_2 = 4.0 \text{ g} \times \frac{1}{1000} \text{ kg/g} = 0.004 \text{ kg} \]

Step 3: Solving for Velocity of 2.0g Cricket

Use the kinetic energy formula to solve for the velocity of the 2.0 g cricket:\[ 4.186 = \frac{1}{2} \cdot 0.002 \cdot v^2 \]First, isolate \( v^2 \):\[ v^2 = \frac{4.186}{0.001} \]\[ v^2 = 4186 \]Thus, the velocity \( v \) is:\[ v = \sqrt{4186} \approx 64.7 \text{ m/s} \]

Step 4: Solving for Velocity of 4.0g Cricket

Use the kinetic energy formula to solve for the velocity of the 4.0 g cricket:\[ 4.186 = \frac{1}{2} \cdot 0.004 \cdot v^2 \]First, isolate \( v^2 \):\[ v^2 = \frac{4.186}{0.002} \]\[ v^2 = 2093 \]Thus, the velocity \( v \) is:\[ v = \sqrt{2093} \approx 45.7 \text{ m/s} \]

Key Concepts

Physics Problems

Physics problems can be challenging, but they're an exciting way to apply mathematical formulas to real-world situations. When solving these problems, it's crucial to begin by understanding the variables and what the problem is asking you to find. In this example, we needed to calculate the velocity of crickets so that their kinetic energy would match a specific amount.

• Identify the problem: Find out what you need to calculate and the given data.
• Use the right formula: Here, we used the formula for kinetic energy, \( KE = \frac{1}{2}mv^2 \) where \( m \) is mass and \( v \) is velocity.
• Unit conversion: Always convert your units into the standard form, such as kilograms for mass.

Key to solving physics problems is understanding these steps and applying them consistently. Make sure to double-check your calculations, especially when dealing with different units.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In our exercise, the focus was on converting kinetic energy to a numerical value using the kinetic energy formula. Kinetic energy is the energy an object possesses because of its motion. Understanding energy conversion can help in recognizing how various energy forms relate to each other.
Kinetic energy is dependent on both the mass and velocity of an object:

• Mass (\( m \)): The amount of matter in the object, affecting how much kinetic energy the object can have.
• Velocity (\( v \)): The speed at which the object is moving, playing a pivotal role since kinetic energy increases with the square of velocity \( (v^2) \).

The formula involves multiplying the mass and the square of the velocity, then halving it. This action shows how energy is converted from its potential form (the cricket poised to jump) to kinetic form (the cricket in flight). Understanding this principle is vital for many physics applications, from observing crickets to understanding massive celestial bodies.

Velocity Calculation

Calculating velocity requires manipulating the kinetic energy formula to solve for \( v \). Let's walk through how to determine the velocity required for our cricket example:
Firstly, convert mass to kilograms to fit the formula's requirements. For instance, convert 2.0 g to 0.002 kg.
Next, substitute the known values into the kinetic energy formula \( KE = \frac{1}{2}mv^2 \). With \( KE \) as 4.186 J and \( m \) as 0.002 kg:

• Rearrange to solve for \( v^2 \): \[ v^2 = \frac{2 \times KE}{m} \]
• Input known values: \[ v^2 = \frac{2 \times 4.186}{0.002} = 4186 \]
• Calculate \( v \) by finding the square root of the result: \[ v = \sqrt{4186} \approx 64.7 \text{ m/s} \]

This calculation allows us to find the cricket's velocity needed for a specific amount of energy. Using mathematical manipulation, we effectively turned energy values into velocity, showcasing how physics allows for the conversion of energy into real-world speed.