Help with relativistic momentum question

[DunWorry]

Homework Statement

a D meson is at rest and decays into a Kaon and a Pion. The Kaon moves with speed 0.867c and has a mass of 0.494 GeV/C^2. The pion has a mass of 0.140 GeV/C^2. use conservation of momentum to calculate the speed of the Pion.

Homework Equations

Relativistic Momentum P = [itex]\gamma[/itex]mV

where [itex]\gamma[/itex] is [itex]\frac{1}{\sqrt{1 -\frac{v^{2}}{c^{2}}}}[/itex]

The Attempt at a Solution

So if the D meson is initally at rest, initial momentum = 0, which means

[itex]\gamma[/itex][itex]_{v1}[/itex]m[itex]_{1}[/itex]v[itex]_{1}[/itex] = [itex]\gamma[/itex][itex]_{v2}[/itex]m[itex]_{2}[/itex]v[itex]_{2}[/itex]

Where particle 1 is the Kaon and particle 2 is the Pion, we want the speed of Pion so we solve for v[itex]_{2}[/itex]

After some rearrangement I got v[itex]_{2}[/itex][itex]^{2}[/itex] = [itex]\frac{1} {\frac{m_{2}^{2}}{(\gamma_{v1}m_{1}v_{1})^{2}} + \frac{1}{c^{2}}}[/itex]

After plugging in the numbers m2[itex]^{2}[/itex] = ([itex]\frac{0.140x10^{9}}{(3x10^{8})^{2}}[/itex])[itex]^{2}[/itex]

and m1[itex]^{2}[/itex] = ([itex]\frac{0.494x10^{9}}{(3x10^{8})^{2}}[/itex])[itex]^{2}[/itex]

and [itex]\gamma[/itex][itex]_{v1}[/itex] = [itex]\frac{1}{\sqrt{1 - 0.867^{2}}}[/itex]
I get an answer faster than light, where have I gone wrong?

 

[mfb]

I would expect that your rearrangement is wrong. Even if you use wrong numbers for the masses, the speed has to be below the speed of light in every relativistic calculation.

Edit: After a closer look at your equation, v₂ calculated there should always be below c. The formula might be right, but then your evaluation is wrong.

 

[vela]

In special relativity problems, you'll find it generally better to stick to working with energy and momentum rather than velocities as it simplifies the algebra quite a bit. Try finding E and p for the pion. Once you have those, you can find its speed using the relation v/c = pc/E.