$^{-1}$ and 2 kg second part moving with a velocity of 8 ms$^{-1}$. What is the mass and velocity of the third part?

To solve this problem, we need to use the law of conservation of momentum. According to this law, the total momentum of a system of objects remains constant if there are no external forces acting on the system. In other words, the sum of the momenta of all the objects before the explosion is equal to the sum of the momenta of all the objects after the explosion.

Before the explosion, the rock had zero velocity, so its momentum was zero. After the explosion, the two parts that went off at right angles to each other have the following momenta:

First part: momentum = mass x velocity = 1 kg x 12 ms$^{-1}$ = 12 kg ms$^{-1}$

Second part: momentum = mass x velocity = 2 kg x 8 ms$^{-1}$ = 16 kg ms$^{-1}$

The total momentum of these two parts is:

Total momentum = 12 kg ms$^{-1}$ + 16 kg ms$^{-1}$ = 28 kg ms$^{-1}$

According to the law of conservation of momentum, the momentum of the third part must be equal and opposite to the total momentum of the first two parts. Let's call the mass of the third part "m" and its velocity "v". Then we have:

Momentum of third part = -28 kg ms$^{-1}$

Momentum = mass x velocity

Therefore:

-mv = -28 kg ms$^{-1}$

Solving for "m", we get:

m = 28/v

Now we can use the law of conservation of energy to find the velocity of the third part. According to this law, the total kinetic energy of a system of objects remains constant if there are no external forces acting on the system. In other words, the sum of the kinetic energies of all the objects before the explosion is equal to the sum of the kinetic energies of all the objects after the explosion.

Before the explosion, the rock had zero kinetic energy, so its total kinetic energy was zero. After the explosion, the two parts that went off at right angles to each other have the following kinetic energies:

First part: KE = 0.5 x mass x velocity$^2$ = 0.5 x 1 kg x (12 ms$^{-1}$)$^2$ = 72 J

Second part: KE = 0.5 x mass x velocity$^2$ = 0.5 x 2 kg x (8 ms$^{-1}$)$^2$ = 64 J

The total kinetic energy of these two parts is:

Total KE = 72 J + 64 J = 136 J

According to the law of conservation of energy, the kinetic energy of the third part must be equal to the difference between the total kinetic energy of the first two parts and the initial kinetic energy of the rock. The initial kinetic energy of the rock was zero, so we have:

KE of third part = Total KE - 0 = 136 J

Using the formula for kinetic energy, we can write:

0.5mv$^2$ = 136 J

Solving for "v", we get:

v = $\sqrt{\frac{272}{m}}$

Substituting