Particle Motion: Retardation & Arithmetic Progression

[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
The answer is -k²*x^-3

(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+x_o , x+2x_o , x+3x_o

 

[rock.freak667]

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.

 

[Cbray]

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

 

[rock.freak667]

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)

 

[asteriatic]

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.