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Chapter 15: Problem 24

\(\bullet\) Suppose some insects have speeds of \(10.00 \mathrm{m} / \mathrm{s}, 8.00 \mathrm{m} / \mathrm{s},\) \(7.00 \mathrm{m} / \mathrm{s},\) and 2.00 \(\mathrm{m} / \mathrm{s} .\) Find (a) the rms speed of these critters and (b) their average speed.

Short Answer

Expert verified
RMS speed is approximately 7.37 m/s, and average speed is 6.75 m/s.

Step by step solution

01

Calculate RMS Speed

The root mean square (RMS) speed is calculated using the formula: \[ v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{n}} \] where \( n \) is the number of measurements. In this case, \( v_1 = 10.00 \, \mathrm{m/s} \), \( v_2 = 8.00 \, \mathrm{m/s} \), \( v_3 = 7.00 \, \mathrm{m/s} \), and \( v_4 = 2.00 \, \mathrm{m/s} \). There are 4 measurements, so \( n = 4 \). Substituting the values, we get:\[ v_{rms} = \sqrt{\frac{10.00^2 + 8.00^2 + 7.00^2 + 2.00^2}{4}} \] \[ v_{rms} = \sqrt{\frac{100.00 + 64.00 + 49.00 + 4.00}{4}} \] \[ v_{rms} = \sqrt{\frac{217.00}{4}} \] \[ v_{rms} = \sqrt{54.25} \] \[ v_{rms} \approx 7.37 \, \mathrm{m/s} \].
02

Calculate Average Speed

The average speed is calculated using the formula: \[ v_{avg} = \frac{v_1 + v_2 + v_3 + v_4}{n} \] Substitute the values we have:\[ v_{avg} = \frac{10.00 + 8.00 + 7.00 + 2.00}{4} \] \[ v_{avg} = \frac{27.00}{4} \] \[ v_{avg} = 6.75 \, \mathrm{m/s} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Speed
Average speed is a fundamental concept when analyzing movement. To find it, we sum up all the individual speeds and divide by the total number of speed values. This provides a central value representing the overall speed.

In practical terms, average speed tells us how fast an object has traveled over a specific distance and period. It is a simple but powerful measure in everyday scenarios, like a car's journey or an athlete's lap time.
  • The formula is straightforward: \[ v_{avg} = \frac{v_1 + v_2 + v_3 + v_4}{n} \]
  • Here, the sum of all speeds is divided by the count of speed values (\( n \)).
Understanding average speed can provide insights into the efficiency and pace of movement in various contexts, simplifying how we evaluate motion.
Velocity Calculation
Where average speed gives an overall rate, velocity adds direction into the mix. It isn鈥檛 just about how fast something is going, but also where it鈥檚 headed.

In physics, calculating velocity involves both speed and direction, making it a vector quantity. Unlike average speed, which only considers magnitude, velocity's directional component is crucial for detailed motion analysis.
  • I used precisely in scenarios such as navigation and engineering.
  • Helps in predicting future positions if the path and time are known.
Think of velocity as a roadmap鈥攑roviding not just speed but also the journey's direction, offering a more complete picture of an object鈥檚 motion.
Kinematics
Kinematics is the study of motion without worrying about the forces causing it. It includes key concepts such as displacement, speed, velocity, and acceleration.

In kinematics, we delve into the trajectory of objects, how they move, and change positions over time. It forms the backbone of how we understand physical movements from simple walks to complex robotic movements.
  • Includes concepts like displacement, velocity, and acceleration.
  • Helps in designing machinery and understanding natural phenomena.
Kinematics provides the framework for analyzing movements in mechanical systems and lays the groundwork for more complex physics concepts.
Root Mean Square
The Root Mean Square (RMS) speed is a statistical measure often used in physics to determine the speed of particles in a system. It's particularly useful in scenarios involving gases or grouped objects moving at various speeds.

RMS provides a kind of average that squares each value, takes the mean, and then the square root, ensuring that all values contribute positively, especially when some are negative or have different signs.
  • The formula for RMS speed: \[ v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{n}} \]
  • Captures fluctuations in speed by focusing on the square of values.
Using RMS can be beneficial for achieving a more accurate representation of speeds, especially in non-uniform datasets, like the speed distribution of particles.

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Most popular questions from this chapter

\(\bullet\) A room with dimensions 7.00 \(\mathrm{m}\) by 8.00 \(\mathrm{m}\) by 2.50 \(\mathrm{m}\) is filled with pure oxygen at \(22.0^{\circ} \mathrm{C}\) and 1.00 atm. The molar mass of oxygen is 32.0 \(\mathrm{g} / \mathrm{mol} .\) (a) How many moles of oxygen are required? (b) What is the mass of this oxygen, in kilograms?

\(\bullet\) A cylinder contains 0.250 mol of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas at a temperature of \(27.0^{\circ} \mathrm{C}\) . The cylinder is provided with a frictionless piston, which maintains a constant pressure of 1.00 atm on the gas. The gas is heated until its temperature increases to \(127.0^{\circ} \mathrm{C}\) . Assume that the \(\mathrm{CO}_{2}\) may be treated as an ideal gas. (a) Draw a \(p V\) diagram of this process. (b) How much work is done by the gas in the process? (c) On what is this work done? (d) What is the change in internal energy of the gas? (e) How much heat was supplied to the gas? (f) How much work would have been done if the pressure had been 0.50 atm?

Near room temperature, how does the internal energy of one mole of a diatomic ideal gas compare to that of one mole of a monatomic ideal gas? A. They have the same internal energy. B. The diatomic gas has 2 times as much internal energy as the monatomic gas. C. The diatomic gas has 2\(/ 3\) times as much internal energy as the monatomic gas. D. The diatomic gas has 3\(/ 2\) times as much internal energy as the monatomic gas. E. The diatomic gas has 5\(/ 3\) times as much internal energy as the monatomic gas.

\(\bullet\) Two moles of an ideal gas are heated at constant pressure from \(T=27^{\circ} \mathrm{C}\) to \(T=107^{\circ} \mathrm{C}\) (a) Draw a \(p V\) diagram for this process. (b) Calculate the work done by the gas.

\(\bullet\) Three moles of an ideal gas are in a rigid cubical box with sides of length 0.200 \(\mathrm{m} .\) (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is \(20.0^{\circ} \mathrm{C}\) ? (b) What is the force when the temperature of the gas is increased to \(100.0^{\circ} \mathrm{C} ?\)
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