A particle of mass M moves through the lab at 0.6c. Suddenly it explodes into two fragments. First

fragment, with mass 2M/3
, moves at 0.8c in the same direction that the original particle had been moving.
Determine the velocity (magnitude and direction) of the second fragment. What is the nature (or name) of
the second particle?

Are your quoted masses (M; 2M/3) rest masses or masses in the laboratory frame of reference?

Momentum must be conserved in the laboratory rest frame as well as that of the incoming particle. Mass does not have to be conserved. (There could have been a conversion of mass to kinetic energy)

I do not see how the particle can be identified from the limited information provided.

To determine the velocity of the second fragment, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion must be equal to the total momentum after the explosion.

Given:
Mass of the original particle (M)
Velocity of the original particle (0.6c)
Mass of the first fragment (2M/3)
Velocity of the first fragment (0.8c) in the same direction as the original particle

Let's assume the mass of the second fragment is "x."

Using the conservation of momentum equation:
Total initial momentum = Total final momentum

(M * 0.6c) = (2M/3) * (0.8c) + (x * vf)

Simplifying and solving for vf:
0.6Mc = (2M/3)(0.8c) + x * vf

0.6Mc - (2M/3)(0.8c) = x * vf

0.6Mc - (1.6M/3)c = x * vf

0.2Mc = x * vf

The velocity of the second fragment (vf) can be found by rearranging the equation:

vf = (0.2Mc) / x

Now, we need to determine the mass (x) of the second fragment.

To do that, we'll use the principle of conservation of mass. According to this principle, the total mass before the explosion must be equal to the total mass after the explosion.

Initial mass = Final mass

M = (2M/3) + x

Simplifying and solving for x:
3M = 2M + 3x

3x = M

x = M/3

Substituting the value of x back into the velocity equation:

vf = (0.2Mc) / (M/3)

vf = 0.6c

Therefore, the velocity of the second fragment is 0.6c in the same direction as the original particle.

The nature (or name) of the second particle is not given in the question, so we cannot determine it based on the information provided.