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- Homework Statement
- Taylor Classical Mechanics 15.84

- Relevant Equations
- Below

Source = John R. Taylor, Classical Mechanics, page 651 + page 677

Trying to solve,

A mass [itex]m[/itex] is thrown from the origin at t=0 with initial three momentum [itex]p_0[/itex] in the y direction. If it is subject to a constant force [itex]F_0[/itex] in the x direction, find its velocity [itex]\mathbf{v}[/itex] as a function of t, and by integrating [itex]\mathbf{v}[/itex] find its trajectory.

Taylor solves this and I slowly worked this problem if mass released from rest.

$$\gamma = \sqrt{1+\bigg(\frac{Ft}{mc}\bigg)^2} $$

$$\mathbf{v}(t)=\frac{\mathbf{p}}{m\gamma}=\frac{\mathbf{F}t}{m\sqrt{1+(Ft/mc)^2}}$$

$$\mathbf{x}(t)=\frac{\mathbf{F}}{m}\left(\frac{mc}{F}\right)^2\left(\sqrt{1+\left(\frac{Ft}{mc}\right)^2}-1\right)$$

I am not sure how I could get this specific.

Thoughts=

There exists [itex] \gamma_0 [/itex] at [itex]t=0[/itex], and evolves to [itex]\gamma[/itex]. I see this [itex]\gamma_0[/itex] altering general case.

Trying to solve,

A mass [itex]m[/itex] is thrown from the origin at t=0 with initial three momentum [itex]p_0[/itex] in the y direction. If it is subject to a constant force [itex]F_0[/itex] in the x direction, find its velocity [itex]\mathbf{v}[/itex] as a function of t, and by integrating [itex]\mathbf{v}[/itex] find its trajectory.

Taylor solves this and I slowly worked this problem if mass released from rest.

$$\gamma = \sqrt{1+\bigg(\frac{Ft}{mc}\bigg)^2} $$

$$\mathbf{v}(t)=\frac{\mathbf{p}}{m\gamma}=\frac{\mathbf{F}t}{m\sqrt{1+(Ft/mc)^2}}$$

$$\mathbf{x}(t)=\frac{\mathbf{F}}{m}\left(\frac{mc}{F}\right)^2\left(\sqrt{1+\left(\frac{Ft}{mc}\right)^2}-1\right)$$

I am not sure how I could get this specific.

Thoughts=

There exists [itex] \gamma_0 [/itex] at [itex]t=0[/itex], and evolves to [itex]\gamma[/itex]. I see this [itex]\gamma_0[/itex] altering general case.