Problem 36 - Elementary Algebra

Question

Use numerical evaluation on the equations. Physics (energy) \(E=\frac{1}{2} m v^{2} . \) Find \(E\) if \(m=12\) and \(v=5\)

Step-by-Step Solution

Answer

Answer: The kinetic energy of the object is 150 Joules.

Step 1: Write down the given values and formula

We have the Kinetic Energy formula: \(E=\frac{1}{2} m v^{2}\) and the given values of mass (m) and velocity (v): - m = 12 kg - v = 5 m/s

Step 2: Substitute the values into the formula

Now, plug in the values of m and v into the formula: \(E=\frac{1}{2} (12) (5)^{2}\)

Step 3: Evaluate the expression

Evaluate the expression step-by-step. First, find the square of the velocity: \((5)^{2} = 25\) Next, multiply the mass and velocity squared: \((12)(25)= 300\) Finally, multiply by 0.5 to get the kinetic energy: \(E= \frac{1}{2}(300)= 150\)

Step 4: Write down the final answer

Kinetic energy E is 150 Joules.

Key Concepts

Numerical Evaluation

When solving physics problems, numerical evaluation is the process of calculating an exact value for a physical quantity. It involves taking the known numerical values of variables and substituting them into an equation to find a numerical answer. It’s like following a recipe; you combine your known ingredients in the right way to create your result.

Steps for Numerical Evaluation:

• Identify the variables and their given numerical values.
• Ensure units are consistent and convert if necessary.
• Substitute the numbers into the equation accurately.
• Perform mathematical operations in the correct order (apply the order of operations, also known as PEMDAS/BODMAS).
• State your answer with the appropriate unit of measurement.

In the case of the kinetic energy problem given, the numerical evaluation starts with squaring the velocity, multiplying it by the mass, and finally multiplying by a half to determine the energy in joules.

Physics Formulas

Physics formulas are the backbone of solving problems in physics. They represent relationships between different physical quantities and enable calculations based on these relationships. To work with them effectively, familiarity with the formulas and a strong foundation in algebra is necessary.

Important Aspects of Physics Formulas:

• Understand what each variable in the formula represents.
• Identify whether the formula requires input variables to be in specific units.
• Know the constraints or conditions under which the formula can be applied.

The kinetic energy formula, \(E=\frac{1}{2}mv^2\), connects mass (m) and velocity (v) to the energy (E) of a moving object. Recognizing that kinetic energy is dependent on the square of the velocity and directly proportional to the mass of the object is critical in understanding how changes in these variables affect the energy.

Substituting Values

Substituting values into an equation is a critical step in the problem-solving process. It involves replacing the variables in an equation with the given numerical figures to allow for computation.

Tips for Accurate Substitution:

• Write the formula down before inserting the numbers to minimize errors.
• Ensure that all the values are in the proper units before substitution.
• Label all values with their respective units for clarity.
• Use parentheses to group terms correctly, especially when using a calculator.

For example, in the kinetic energy equation, after noting that the mass is 12 kg and the velocity is 5 m/s, these values are substituted directly into the equation in place of the mass and velocity variables. This process yields the numerical expression which can then be evaluated.

Solving Kinetic Energy Problems

Kinetic energy problems often involve determining the energy of motion possessed by an object. Solving these requires an understanding of the kinetic energy formula and the ability to accurately manipulate it. Students should also be aware of the forms energy can take and the principle of energy conservation.

Key Aspects of Solving Kinetic Energy Problems:

• Start by writing down what is given and what is required.
• Choose the correct formula to apply, such as the kinetic energy formula for moving objects.
• Understand the physical meaning behind the variables and the answer.
• Execute proper algebraic skills to isolate the desired variable if necessary.
• Review your final answer to see if it is reasonable in the context of the problem.

In the exercise presented, kinetic energy is calculated by substituting the given values of mass and velocity into the kinetic energy equation. The steps involve mathematical operations such as squaring and multiplication, leading to the answer with the correct unit, joules, indicative of energy.