Problem 38
Question
Use numerical evaluation on the equations. Physics (energy) \(E=\frac{1}{2} m v^{2} . \) Find \(E\) if \(m=24.02\) and \(v=7\)
Step-by-Step Solution
Verified Answer
Question: Calculate the kinetic energy of an object with a mass of 24.02 kg and a velocity of 7 m/s.
Answer: The kinetic energy of the object is 588.98 Joules (J).
1Step 1: Write down the given values
Mass (m) = 24.02 kg
Velocity (v) = 7 m/s
2Step 2: Write down the formula for kinetic energy
Kinetic Energy (E) = \(\frac{1}{2}mv^2\)
3Step 3: Substitute the given values into the formula
Replace m with 24.02 kg and v with 7 m/s in the formula:
E = \(\frac{1}{2}(24.02\,\text{kg})(7\,\text{m/s})^2\)
4Step 4: Perform the calculations
First, square the velocity (7 m/s):
\((7\,\text{m/s})^2 = 49\,\text{m}^2/\text{s}^2\)
Then, multiply the squared velocity by the mass and divide by 2:
E = \(\frac{1}{2}(24.02\,\text{kg})(49\,\text{m}^2/\text{s}^2) = 588.98\,\text{J}\)
5Step 5: Write down the final result
The kinetic energy (E) of the object is 588.98 Joules (J).
Key Concepts
Numerical EvaluationPhysics EquationsMass and Velocity Calculations
Numerical Evaluation
Numerical evaluation is a simple concept that involves calculating values by substituting numbers into equations. When solving physics problems like calculating kinetic energy, it's all about taking the equation and inserting the given numerical values for mass and velocity. This practical approach helps simplify the raw mathematical expressions into meaningful results.
For instance, in this exercise, we have been given the values for mass (24.02 kg) and velocity (7 m/s). By using numerical evaluation, we substitute these values into the kinetic energy formula to find the exact energy. This method is key in physics because it allows us to turn theoretical physics equations into real-world answers.
By performing each calculation step-by-step, such as squaring the velocity and inserting it into the equation, numerical evaluation ensures a clear understanding and accurate results.
For instance, in this exercise, we have been given the values for mass (24.02 kg) and velocity (7 m/s). By using numerical evaluation, we substitute these values into the kinetic energy formula to find the exact energy. This method is key in physics because it allows us to turn theoretical physics equations into real-world answers.
By performing each calculation step-by-step, such as squaring the velocity and inserting it into the equation, numerical evaluation ensures a clear understanding and accurate results.
Physics Equations
Physics equations are formulae that express the relationships between different physical quantities. They represent the natural laws and principles governing the physical world. In this case, the equation for kinetic energy, \( E = \frac{1}{2} mv^2 \), represents the energy an object possesses due to its motion.
The energy equation shows that kinetic energy depends on both the object's mass (\( m \)) and the square of its velocity (\( v^2 \)). The multiplicative constant \( \frac{1}{2} \) is derived from the work-energy theorem and ensures that the units of energy remain consistent with the measurable values of mass and velocity.
Understanding physics equations enables students to interpret scientific phenomena and predict outcomes based on specific variables. When students encounter such equations, they should focus on identifying each component's role and how altering any value would affect the outcome, like how increasing velocity disproportionately increases kinetic energy.
The energy equation shows that kinetic energy depends on both the object's mass (\( m \)) and the square of its velocity (\( v^2 \)). The multiplicative constant \( \frac{1}{2} \) is derived from the work-energy theorem and ensures that the units of energy remain consistent with the measurable values of mass and velocity.
Understanding physics equations enables students to interpret scientific phenomena and predict outcomes based on specific variables. When students encounter such equations, they should focus on identifying each component's role and how altering any value would affect the outcome, like how increasing velocity disproportionately increases kinetic energy.
Mass and Velocity Calculations
Mass and velocity calculations are fundamental to problems related to motion and energy. Mass refers to the amount of matter present in an object, and it is usually measured in kilograms (kg). Velocity, on the other hand, is the rate of change of an object's position with respect to time, measured in meters per second (m/s). Both of these quantities are crucial in calculating kinetic energy.
In the given exercise, to find the kinetic energy, you start by calculating the velocity squared. Squaring the velocity increases its influence on the energy significantly because small changes in velocity result in large changes in kinetic energy.
Once velocity is squared, this value is multiplied by the mass to ensure the energy is in the correct units, Joules (J). Finally, the product is halved, as per the kinetic energy formula. This sequence of steps highlights the delicate balance between mass and velocity, demonstrating their combined effect on the kinetic energy of an object. Understanding and correctly executing these calculations are key skills in physics.
In the given exercise, to find the kinetic energy, you start by calculating the velocity squared. Squaring the velocity increases its influence on the energy significantly because small changes in velocity result in large changes in kinetic energy.
Once velocity is squared, this value is multiplied by the mass to ensure the energy is in the correct units, Joules (J). Finally, the product is halved, as per the kinetic energy formula. This sequence of steps highlights the delicate balance between mass and velocity, demonstrating their combined effect on the kinetic energy of an object. Understanding and correctly executing these calculations are key skills in physics.
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Problem 38
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