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Chapter 5: Problem 6

The average speed of a nitrogen molecule in air is about \(6.70 \times 10^{2} \mathrm{~m} / \mathrm{~s},\) and its mass is \(4.68 \times 10^{-26} \mathrm{~kg} .\) (a) If it takes \(3.00 \times 10^{-13}\) s for a nitrogen molecule to hit a wall and rebound with the same speed but moving in the opposite direction, what is the average acceleration of the molecule during this time interval? (b) What average force does the molecule exert on the wall?

Short Answer

Expert verified
The average acceleration of the nitrogen molecule during this time interval is \(-4.47 \times 10^{15}\ m/s^2\) and the average force it exerts on the wall is \(-2.09 \times 10^{-10}\ N\).

Step by step solution

01

Calculate the change in velocity.

Before the nitrogen molecule hits the wall, its velocity is \(6.70 \times 10^{2}\ m/s\). After it rebounds, it's moving at the same speed, but in the opposite direction, so its new velocity is \(-6.70 \times 10^{2}\ m/s\). Therefore, the change in velocity is \((-6.70 \times 10^{2} m/s) - (6.70 \times 10^{2} m/s) = -1.34 \times 10^{3}\ m/s\).
02

Calculate the average acceleration.

To calculate the average acceleration, we use the formula \(\Delta v / \Delta t\). Substituting the given values we get \((-1.34 \times 10^{3}\ m/s) / (3.00 \times 10^{-13}\ s) = -4.47 \times 10^{15}\ m/s^2\). This is the average acceleration of the molecule during the given time interval.
03

Calculate the average force exerted by the nitrogen molecule.

The force exerted by the molecule can be calculated using the formula \(F = ma\). Substituting the given values, we get \(F = (4.68 \times 10^{-26}\ kg)*(-4.47 \times 10^{15}\ m/s^2) = -2.09 \times 10^{-10}\ N\). The negative sign indicates that the direction of the force is opposite to the initial direction of the molecule.

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Key Concepts

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

Average Speed
The concept of average speed is crucial in understanding the movement of particles. Average speed is defined as the total distance traveled divided by the total time taken to travel that distance.
  • Mathematically, it is expressed as: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
  • It provides a simplified overview, representing how fast an object is traveling on average.
  • In the case of the nitrogen molecule in our exercise, the average speed is given as \(6.70 \times 10^{2} \text{ m/s}\).
Average speed does not account for variations in speed that might occur over different segments of the journey. This is important to keep in mind when solving physics problems, as average speed only considers the overall route taken, not the specifics of each segment.
Acceleration
Acceleration is a fundamental concept in kinematics, describing how quickly an object's velocity changes over time. In the exercise, we calculated the average acceleration using the change in velocity and the time interval.
  • Acceleration can be calculated using the formula: \[ a = \frac{\Delta v}{\Delta t} \]
  • Where \( \Delta v \) is the change in velocity and \( \Delta t \) is the time taken for that change.
  • For the nitrogen molecule, the change in velocity was \( -1.34 \times 10^{3} \text{ m/s}\), and the time interval was \(3.00 \times 10^{-13} \text{ s}\).
  • This resulted in an average acceleration of \(-4.47 \times 10^{15} \text{ m/s}^2\).
Acceleration is a vector quantity, meaning it has both magnitude and direction. In this problem, the negative acceleration indicates the direction is opposite to the initial motion of the nitrogen molecule.
Force Exerted by a Particle
Understanding the force exerted by a particle involves Newton's second law of motion, which states that force is the product of mass and acceleration. This principle is used to determine the force exerted by the nitrogen molecule against the wall.
  • The formula for force is: \[ F = ma \]
  • In the exercise, the force was calculated using the mass of the molecule (\(4.68 \times 10^{-26} \text{ kg}\)) and its average acceleration (\(-4.47 \times 10^{15} \text{ m/s}^2\)).
  • The average force exerted was \(-2.09 \times 10^{-10} \text{ N}\).
The negative sign in the result indicates that the force direction is opposite to the initial velocity of the molecule. This is crucial for correctly interpreting the physical interaction between the molecule and the wall. The concept of force is essential in understanding how objects interact with each other in terms of pushing or pulling actions.
Kinetic Theory of Gases
The kinetic theory of gases offers insights into the behavior of gases at the molecular level. It helps explain properties such as pressure, temperature, and volume based on the assumptions that gases are composed of a large number of small particles in constant, random motion.
  • This theory posits that gas molecules are constantly in motion, and their speed can be used to predict the temperature of the gas.
  • The pressure exerted by a gas is due to collisions of gas molecules with the walls of its container, as seen with the nitrogen molecule hitting the wall in our exercise.
  • The average kinetic energy of gas molecules is directly proportional to the temperature of the gas.
These core ideas of the kinetic theory provide the foundational understanding necessary to predict the behavior of gases under different conditions. It bridges the gap between macroscopic properties and microscopic activities.

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Most popular questions from this chapter

A block weighing \(75.0 \mathrm{N}\) rests on a plane inclined at \(25.0^{\circ}\) to the horizontal. A force \(F\) is applied to the object at \(40.0^{\circ}\) to the horizontal, pushing it upward on the plane. The coefficients of static and kinetic friction between the block and the plane are, respectively, 0.363 and \(0.156 .\) (a) What is the minimum value of \(F\) that will prevent the block from slipping down the plane? (b) What is the minimum value of \(F\) that will start the block moving up the plane? (c) What value of \(F\) will move the block up the plane with constant velocity?

To prevent a box from sliding down an inclined plane, student A pushes on the box in the direction parallel to the incline, just hard enough to hold the box stationary. In an identical situation student \(B\) pushes on the box horizontally. Regard as known the mass \(m\) of the box, the coefficient of static friction \(\mu_{s}\) between box and incline, and the inclination angle \(\theta\). (a) Determine the force \(A\) has to exert. (b) Determine the force \(B\) has to exert. (c) If \(m=2.00 \mathrm{kg}, \theta=25.0^{\circ},\) and \(\mu_{s}=0.160,\) who has the easier job? (d) What if \(\mu_{s}=0.380 ?\) Whose job is easier?

One side of the roof of a building slopes up at \(37.0^{\circ} .\) A student throws a Frisbee onto the roof. It strikes with a speed of \(15.0 \mathrm{m} / \mathrm{s}\) and does not bounce, but slides straight up the incline. The coefficient of kinetic friction between the plastic and the roof is \(0.400 .\) The Frisbee slides \(10.0 \mathrm{m}\) up the roof to its peak, where it goes into free fall, following a parabolic trajectory with negligible air resistance. Determine the maximum height the Frisbee reaches above the point where it struck the roof.

An \(8.40-\mathrm{kg}\) object slides down a fixed, frictionless inclined plane. Use a computer to determine and tabulate the normal force exerted on the object and its acceleration for a series of incline angles (measured from the horizontal) ranging from \(0^{\circ}\) to \(90^{\circ}\) in \(5^{\circ}\) increments. Plot a graph of the normal force and the acceleration as functions of the incline angle. In the limiting cases of \(0^{\circ}\) and \(90^{\circ},\) are your results consistent with the known behavior?

You stand on the seat of a chair and then hop off. (a) During the time you are in flight down to the floor, the Earth is lurching up toward you with an acceleration of what order of magnitude? In your solution explain your logic. Model the Earth as a perfectly solid object. (b) The Earth moves up through a distance of what order of magnitude?
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