I've been having a lot of trouble with this problem. There's definitely something I'm missing and it most likely has to do with the force.(adsbygoogle = window.adsbygoogle || []).push({});

"A mass [tex]m[/tex] is thrown from the origin at [tex]t = 0[/tex] with initial three-momentum [tex]p_0[/tex] in the y direction. If it is subject to a constant force [tex]F_0[/tex] in the x direction, find its velocity [tex]v[/tex] as a function of [tex]t[/tex] and by integrating [tex]v[/tex], find its trajectory. You will need to integrate functions such as [tex]t/\sqrt{a + bt^2}[/tex] and [tex]1/\sqrt{a + bt^2}[/tex].

In addition, check that in the non-relativistic limit, ([tex]c \rightarrow \infty[/tex]), [tex]x(t)[/tex] is what you expect for motion in a constant field and check that [tex]y(t)[/tex] is what you expect for motion in a constant field when the force is orthogonal to the y direction.

HINT: You will need the Taylor expansion of the functions [tex]\sqrt{1 + x}[/tex] and [tex]\ln(1 + x)[/tex]."

The 2nd part seems easy, but I'm simply not sure how to find [tex]x(t)[/tex] or [tex]y(t)[/tex] in the first place.

My work so far:

[tex]\gamma = 1/\sqrt(1 - v^2/c^2)\\

\gamma_v_0 = 1/\sqrt(1 - {v_0}^2/c^2)[/tex]

[tex]p_0 = (0, p_0, 0) = m*\gamma_v0*(0, v_0, 0)[/tex]

[tex]F_0 = (F_0, 0, 0)[/tex]

[tex]F = dP/dt, \mbox{so } P - p_0 = F*t[/tex]

[tex]P = F_0*t + p_0 = m*\gamma*v[/tex]

[tex]v = (v_x, v_y, v_z), v_z = 0[/tex]

[tex]m*\gamma*v_x = F_0*t[/tex]

[tex]m*\gamma*v_y = p_0 = m*\gamma_v0*v_0[/tex]

I'm not exactly sure how to proceed from here since I can't really isolate [tex]v_x[/tex] or [tex]v_y[/tex] because the gamma term contains only the magnitude of [tex]v[/tex]. Should I use [tex]\|v\| = \sqrt{v_x^2 + v_y^2}[/tex] and then work through some really terrible algebra? Or is this even the right way to approach this problem?

Edit: I've actually done the terrible alegbra using [tex]\|v\| = \sqrt{v_x^2 + v_y^2}[/tex], but the equations I end up with are ridiculous. Can anyone suggest a different method?

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# Homework Help: Relativistic Force and Momentum

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