# Particle Motion: Retardation & Arithmetic Progression

• Cbray
In summary, a particle moves away from a fixed point O in a straight line, with its speed given by v=k/x for some constant k. Its retardation is inversely proportional to x^3. Additionally, if points a,b,c,d are in order on the straight line with equal distances between them, the times taken to travel these distances increase in arithmetic progression. To find the retardation, use the chain rule to express 'a' in terms of 'v' and 'x'. To find the time taken, use the equation t=distance/velocity at the given points.
Cbray

## Homework Statement

A particle moves in a straight line away from a fixed point O in the line, such that when its distance from O is x its speed v is given by v=k/x , for some constant k.

(a) show that the particle has a retardation which is inversely proportional to x3

(b) if a,b,c,d are points in that order on the straight line, such that the distances ab, bc, cd are all equal, show that the times taken to travel these successive distances increase in arithmetic progression.

## Homework Equations

Possible the answer to the first one comes into relevance, -k2*x-3

## The Attempt at a Solution

let the four points have x coordinates
x, x+xo , x+2xo , x+3xo

Last edited:
For part a) how would you find the retardation ? (negative acceleration)

(Hint: use the chain rule to express 'a' in terms of 'v' and 'x')

For part b) try finding time based on t= distance/velocity at the given points.

rock.freak667 said:
For part a) how would you find the retardation ? (negative acceleration)

(Hint: use the chain rule to express 'a' in terms of 'v' and 'x')

For part b) try finding time based on t= distance/velocity at the given points.

I still can't figure it out :L

Cbray said:
I still can't figure it out :L

If a=dv/dt, can you use the chain rule to re-write this in terms of v and x? (Hint: you will have to have a dv/dx term)

where xo is the equal distance between each point

speed v at each point can be written as
v = k/x
v = k/(x+xo)
v = k/(x+2xo)
v = k/(x+3xo)

Using the given equation for speed, we can equate them to find the time taken to travel each distance:
k/x = k/(x+xo) * t1
k/(x+xo) = k/(x+2xo) * t2
k/(x+2xo) = k/(x+3xo) * t3

Simplifying, we get:
t1 = (x+xo)/x
t2 = (x+2xo)/(x+xo)
t3 = (x+3xo)/(x+2xo)

We can see that the difference between t2 and t1 is xo/x, and the difference between t3 and t2 is also xo/x. This shows that the times taken to travel successive distances are increasing by a constant value, which is the definition of an arithmetic progression. Therefore, the times taken to travel distances ab, bc, and cd are in arithmetic progression. Hence, the statement is proven.

## 1. What is particle motion?

Particle motion refers to the movement of individual particles, such as atoms or molecules, in a fluid or gas. It can also refer to the movement of subatomic particles in a vacuum.

## 2. What is retardation in particle motion?

Retardation in particle motion refers to the decrease in speed or deceleration of a particle as it moves through a medium. This can be caused by forces such as friction or air resistance.

## 3. How is arithmetic progression related to particle motion?

Arithmetic progression, also known as arithmetic sequence, is a mathematical concept that involves a sequence of numbers where the difference between each consecutive term is constant. In particle motion, arithmetic progression can be used to analyze the change in position or velocity of a particle over time.

## 4. What factors can affect particle motion?

Some factors that can affect particle motion include the type of medium the particle is moving through, the size and shape of the particle, and any external forces acting on the particle.

## 5. How is particle motion studied and measured in scientific research?

Particle motion can be studied and measured using various scientific tools and techniques, such as high-speed cameras, motion sensors, and computer simulations. These methods allow scientists to track the movement of particles and analyze their behavior under different conditions.

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