An unknown object with mass M slides on a smooth frictionless surface with a speed of 35.0 m/s along the x-axis. Suddenly it explodes into two pieces, P and Q, which move in different directions as shown in the figure below. Determine the final velocity of both fragments if mass of piece Q is one quarter of the original mass.

To determine the final velocity of both fragments, we can use the law of conservation of momentum. According to this law, the total momentum before the explosion is equal to the total momentum after the explosion.

The momentum of an object is equal to the product of its mass and velocity. So, if we denote the mass of the unknown object as M and its initial velocity as V, the initial momentum of the object (and hence the total momentum before the explosion) is given by:

Initial momentum = M * V

After the explosion, the object breaks into two pieces: P and Q. Let's denote the mass of piece Q as mQ and its final velocity as vQ. Given that the mass of piece Q is one quarter of the original mass (mQ = 1/4 * M), we can write its final momentum as:

Final momentum of Q = mQ * vQ

Since momentum is conserved, the total momentum before the explosion is equal to the total momentum after the explosion. Therefore, we have:

Initial momentum = Final momentum of P + Final momentum of Q

(M * V) = Final momentum of P + (mQ * vQ)

Since piece P is not mentioned in the question, we don't have any information about it. However, we can still determine the final velocity of piece Q using the given information.

To find vQ, we need to isolate it in the above equation:

Final momentum of Q = (M/4) * vQ

(M * V) - (M/4) * vQ = Final momentum of P

Now, we need to substitute the given values into the equation. Let's assume the mass of the unknown object is M = 1 kg, its initial velocity V = 35.0 m/s, and the mass of piece Q is mQ = 1/4 * M = 1/4 kg.

(M * V) - (M/4) * vQ = Final momentum of P

(1 kg * 35.0 m/s) - (1 kg/4) * vQ = Final momentum of P

35 kg·m/s - 0.25 kg·vQ = Final momentum of P

Since we don't have any information about piece P, the equation cannot be solved to determine its final momentum or velocity. However, we can determine the relationship between the final velocities of piece P and Q. Let's label the final velocity of piece P as vP.

Final momentum of Q = (M/4) * vQ

(1 kg/4) * vQ = (1/4) * vQ

We can see that the magnitudes of the final momenta of pieces P and Q are equal. Therefore, we can conclude that the magnitudes of their final velocities are also equal:

|vP| = |vQ|

In other words, the final velocity of piece Q is the same as the final velocity of piece P, but they move in opposite directions.

So, based on the given information, we can say that the final velocity of both fragments is 35.0 m/s, with piece Q moving in the opposite direction of the initial motion along the x-axis.