Persistence of Relationships in Inertial Frame R

[imy786]

Homework Statement

An inertial frame R in which the particles’ positions and velocities are related by

A1= - m2 (A2) / m1


V1 = - m2(V2) / m1

at time t = 0. Show that these relationships persist at all subsequent times.

Homework Equations

The Attempt at a Solution

dont know how to start

 

[Andrew Mason]

Homework Statement

An inertial frame R in which the particles’ positions and velocities are related by

A1= - m2 (A2) / m1


V1 = - m2(V2) / m1

at time t = 0. Show that these relationships persist at all subsequent times.

There are no external forces acting on the particles. You can apply Newton's Third law to each particle. There are two possibilities to analyse: 1. there is no change in particle 1 speed and 2. there is a change.

If there is no change in V1, the solution is trivial: V2 cannot change. What does this say about A1 and A2? (dA1/dt = V1)

If particle 1 experiences a change in velocity dV1, what change in velocity must particle 2 experience (Newton's third law)? After the change, what is the relationship between V1 and V2? Between A1 and A2?

AM

 

[imy786]

well if V1 is constant then A2=0
if V2 also remains constant then A2 = 0

is this right...
and how do i shows that these relationships persist all subsequaent times?

 

[Andrew Mason]

well if V1 is constant then A2=0
if V2 also remains constant then A2 = 0

I think you meant "if V1 = constant then A1 = 0.

We are assuming that the only two bodies here are M1 and M2. This means that if there is a force on M1 there must be an equal and opposite force on M2 (Newton's third law), so F1 = M1A1 = - F2 = -M2A2, ==> (1): A1 = -M2A2/M1

If V1 is constant, then A1 =0. If A1 = 0 then from (1): A2 = 0 and V2 is constant. Thus if V1 = M2V2/M1 at time t and V1 is the same for all time and V2 is the same for all time, then the equation is true for all time.

Now the trickier one is to show that if V1 is not constant for a time dt, that the relationship V1 + dV1 = M2(V2+dV2)/M1 holds after time dt. What is the relationship between dV1, A1 and dt? Between dV2, A2 and dt? Are the dt the same for M1 and M2? How do we know (think Newton's laws).

AM