# Central-Force Motion Problem

• sams
In summary, the conversation discusses a problem in Chapter 8 of the Classical Dynamics of Particles and Systems book, where the task is to show that two particles will collide after a specific time. The speaker has no trouble with the derivations and integrations but is unsure why a negative sign was included in Equation (6) and suggests that it should have been in Equation (5) instead. The other person explains that the negative sign is necessary in Equation (4) due to the physical reasons involving the negative velocity of the particles. The conversation ends with gratitude for the help and support.

#### sams

Gold Member
In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 323, Problem 8-5, we are asked to show that the two particles will collide after a time ##\tau/4√2##.

I don't have any problems with the derivations and with the integrations, but I want to know please why the authors put a negative sign in Equation (6) and what do they mean that the negative sign was included due to the fact that the time increases as the distance decreases?

Thanks a lot for your help...

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In my opinion the negative sign should have appeared in equation (5). Equation (4) for ##\dot{x}^2## has two solutions. In this case, the negative square root is required as ##\dot{x}## is negative, for the physical reason given

sams
Thank you so much @PeroK for your continuous support...

## 1. What is central-force motion problem?

The central-force motion problem is a type of motion problem in which a single particle moves under the influence of a central force that is directed towards or away from a fixed point. This type of motion can be seen in many real-world scenarios, such as the orbit of planets around the sun or the motion of electrons around an atomic nucleus.

## 2. What are the key factors that influence central-force motion?

The key factors that influence central-force motion are the magnitude and direction of the central force, the initial position and velocity of the particle, and the mass of the particle. These factors determine the path and speed of the particle as it moves under the influence of the central force.

## 3. How do you calculate the trajectory of a particle in central-force motion?

The trajectory of a particle in central-force motion can be calculated using Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. By solving for the acceleration and integrating with respect to time, the particle's position, velocity, and acceleration can be determined at any given time.

## 4. What is the significance of angular momentum in central-force motion?

Angular momentum is a conserved quantity in central-force motion, meaning that it remains constant throughout the motion. This means that as the particle moves closer to the center of the central force, its velocity increases to maintain a constant angular momentum. This concept is important in understanding the behavior of objects in orbit, as well as other central-force motion scenarios.

## 5. How does central-force motion differ from other types of motion?

Central-force motion differs from other types of motion in that it is restricted to a specific type of force (a central force) and a single particle. In other types of motion, such as projectile motion or circular motion, the force may vary or there may be multiple particles involved. Central-force motion also has unique characteristics, such as the conservation of angular momentum, that make it distinct from other types of motion.