From an inertial refernce point, how to tell if a particle is free?

[catstevens]

Homework Statement

From an inertial reference frame S, the vector position of a particle of mass
m1 = 1kg is given by r1(t)=(tx-hat - t^2y-hat)m.
The vector position of a particle m2=2m1 is given by r2=(t)=(tx-hat +t^3y-hat)m

Is particle #1 free along the y-direction? Explain

Homework Equations

If the particle is free along the y-direction then the y's would equal to 0.

The Attempt at a Solution

r1= -t^2y
r2=+t^3y
=not free?
(I am actually unsure if my definition of free is true)

 

[jbsteele]

No, particle #1 is not free along the y-direction. This is because the y components of the two vector positions are not equal to zero.

 

[thomate1]

Your definition of free is not entirely accurate. In order to determine if a particle is free along a certain direction, we need to look at its acceleration. From an inertial reference frame, a particle is considered free if it is not subject to any external forces. This means that its acceleration along the y-direction should be equal to 0.

In the given scenario, we can calculate the acceleration of particle #1 along the y-direction by taking the second derivative of its position with respect to time. This gives us a value of -2 m/s^2. Since this is not equal to 0, we can conclude that particle #1 is not free along the y-direction.

On the other hand, we can also calculate the acceleration of particle #2 along the y-direction, which gives us a value of 3 m/s^2. This is also not equal to 0, so we can say that particle #2 is also not free along the y-direction.

In summary, from an inertial reference frame, we can determine if a particle is free along a certain direction by looking at its acceleration in that direction. If the acceleration is equal to 0, then the particle is considered free. Otherwise, it is not free.