Question

Suppose particles begin moving in one dimension away from the origin at$\mathit{t}\mathbf{=}\mathbf{0}\mathbf{}$with the following velocities: $\mathbf{0}\mathbf{,}\mathbf{\pm }\mathbf{1}\mathbf{,}\mathbf{\pm }\mathbf{2}\mathbf{,}\mathbf{\pm }\mathbf{3}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}\mathbf{s}$and so on. (a) After 1 s , how will the velocities of the particles depend on distance from the origin? (b) Now consider an observer on one of the moving particles not at the origin. How will the relative velocities of the other particles depend on distance from the observer?

Short answer

(a) The distance will have the same magnitude as the velocities which are 0 m ,1 m ,2 m and 3 m respectively which shows that the velocity is totally dependent on distance. If the time changes, the distance will automatically change.

(b)If there is a different frame, then also the distance will be the same according to the relative velocity.If the time changes, the distance will automatically change.

Step-by-step solution

Identification of given data

The given data can be listed below as:

• The first velocity of the particle is 0 .
• The second velocity of the particle is $\pm 1\hspace{0.33em}\mathrm{m}/\mathrm{s}$.
• The third velocity of the particle is$\pm 2\hspace{0.33em}\mathrm{m}/\mathrm{s}$ .
• The fourth velocity of the particle is $\pm 3\hspace{0.33em}\mathrm{m}/\mathrm{s}$.
• The time taken by the observer is $\mathrm{t}=1\hspace{0.33em}\mathrm{s}.$

Significance of the velocity

Thevelocity is described as the distance a particular object covers in a unit time. The velocity is also directly proportional to the acceleration of an object.

(a) Determination of the dependence of velocity

As the particle’s velocities are relative to mainly one reference frame which is say S frame, then the distance will equal the product of the velocity and the time taken. As the time taken is$\mathrm{t}=1\mathrm{s}$ , then the distance will be equal to the velocity such as role="math" localid="1659175841147" $0\hspace{0.33em}\mathrm{m},1\hspace{0.33em}\mathrm{m},2\hspace{0.33em}\mathrm{m}$and $3\hspace{0.33em}\mathrm{m}.$.

Thus, the distance will have the same magnitude as the velocities which are , $0\hspace{0.33em}\mathrm{m},1\hspace{0.33em}\mathrm{m},2\hspace{0.33em}\mathrm{m}$, and 3 m respectively which shows that the velocity is totally dependent on distance. If the time changes, the distance will automatically change.

(b) Determination of the relative velocity  

In a different frame S' which moves with a different velocity, the velocity that is measured in the different frame is the same as the previous velocity. For an observer in the frame , the whole frame moves with the same velocity. Hence, the particle’s velocity in the previous frame S is the same as the new frame and that shows that the magnitude of the relative velocity is the same at different reference frames.

Thus, if there is a different frame, then also the distance will be the same according to the relative velocity. If the time changes, the distance will automatically change.