- #1
yang32366
- 2
- 0
- 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.