Jibobo
Oct5-05, 09:44 PM
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.
"A mass m is thrown from the origin at t = 0 with initial three-momentum p_0 in the y direction. If it is subject to a constant force F_0 in the x direction, find its velocity v as a function of t and by integrating v, find its trajectory. You will need to integrate functions such as t/\sqrt{a + bt^2} and 1/\sqrt{a + bt^2}.
In addition, check that in the non-relativistic limit, (c \rightarrow \infty), x(t) is what you expect for motion in a constant field and check that y(t) 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 \sqrt{1 + x} and \ln(1 + x)."
The 2nd part seems easy, but I'm simply not sure how to find x(t) or y(t) in the first place.
My work so far:
\gamma = 1/\sqrt(1 - v^2/c^2)\\
\gamma_v_0 = 1/\sqrt(1 - {v_0}^2/c^2)
p_0 = (0, p_0, 0) = m*\gamma_v0*(0, v_0, 0)
F_0 = (F_0, 0, 0)
F = dP/dt, \mbox{so } P - p_0 = F*t
P = F_0*t + p_0 = m*\gamma*v
v = (v_x, v_y, v_z), v_z = 0
m*\gamma*v_x = F_0*t
m*\gamma*v_y = p_0 = m*\gamma_v0*v_0
I'm not exactly sure how to proceed from here since I can't really isolate v_x or v_y because the gamma term contains only the magnitude of v. Should I use \|v\| = \sqrt{v_x^2 + v_y^2} 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 \|v\| = \sqrt{v_x^2 + v_y^2}, but the equations I end up with are ridiculous. Can anyone suggest a different method?
"A mass m is thrown from the origin at t = 0 with initial three-momentum p_0 in the y direction. If it is subject to a constant force F_0 in the x direction, find its velocity v as a function of t and by integrating v, find its trajectory. You will need to integrate functions such as t/\sqrt{a + bt^2} and 1/\sqrt{a + bt^2}.
In addition, check that in the non-relativistic limit, (c \rightarrow \infty), x(t) is what you expect for motion in a constant field and check that y(t) 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 \sqrt{1 + x} and \ln(1 + x)."
The 2nd part seems easy, but I'm simply not sure how to find x(t) or y(t) in the first place.
My work so far:
\gamma = 1/\sqrt(1 - v^2/c^2)\\
\gamma_v_0 = 1/\sqrt(1 - {v_0}^2/c^2)
p_0 = (0, p_0, 0) = m*\gamma_v0*(0, v_0, 0)
F_0 = (F_0, 0, 0)
F = dP/dt, \mbox{so } P - p_0 = F*t
P = F_0*t + p_0 = m*\gamma*v
v = (v_x, v_y, v_z), v_z = 0
m*\gamma*v_x = F_0*t
m*\gamma*v_y = p_0 = m*\gamma_v0*v_0
I'm not exactly sure how to proceed from here since I can't really isolate v_x or v_y because the gamma term contains only the magnitude of v. Should I use \|v\| = \sqrt{v_x^2 + v_y^2} 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 \|v\| = \sqrt{v_x^2 + v_y^2}, but the equations I end up with are ridiculous. Can anyone suggest a different method?