A blue puck with a mass of 3.90×10−2 , sliding with a speed of 0.170 on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass , initially at rest. After the collision, the velocity of the blue puck is 5.0×10−2 in the same direction as its initial velocity.

Find the magnitude of the velocity of the red puck after the collision?

You will have to use conservation of energy, and momentum, having two equations and two unknowns. The algebra is somewhat messy, but at least it is restriced to the x direction.

To find the magnitude of the velocity of the red puck after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision, the blue puck is moving with a velocity of 0.170 in the same direction, and the red puck is initially at rest. The total momentum before the collision is given by:

Total momentum before = (mass of blue puck) * (velocity of blue puck) + (mass of red puck) * (velocity of red puck)

Since the red puck is initially at rest, its velocity is 0. Therefore, the total momentum before the collision simplifies to:

Total momentum before = (mass of blue puck) * (velocity of blue puck)

After the collision, the blue puck is moving with a velocity of 5.0×10−2 in the same direction, and we want to find the magnitude of the velocity of the red puck, which we'll call Vr.

Total momentum after = (mass of blue puck) * (velocity of blue puck) + (mass of red puck) * (velocity of red puck)

Since the blue puck is moving in the same direction as its initial velocity, we can write the equation as:

Total momentum after = (mass of blue puck) * (velocity of blue puck) + (mass of red puck) * (Vr)

Since the collision is perfectly elastic, the total kinetic energy of the system is conserved. Therefore, we can write the equation for the total kinetic energy before and after the collision:

Total kinetic energy before = 0.5 * (mass of blue puck) * (velocity of blue puck)^2
Total kinetic energy after = 0.5 * (mass of blue puck) * (velocity of blue puck)^2 + 0.5 * (mass of red puck) * (Vr)^2

Now, using the principle of conservation of momentum and conservation of kinetic energy, we can set up the following equations:

(mass of blue puck) * (velocity of blue puck) = (mass of blue puck) * (velocity of blue puck) + (mass of red puck) * (Vr)
0.5 * (mass of blue puck) * (velocity of blue puck)^2 = 0.5 * (mass of blue puck) * (velocity of blue puck)^2 + 0.5 * (mass of red puck) * (Vr)^2

Simplifying the equations, we have:

(mass of red puck) * (Vr) = 0
0.5 * (mass of red puck) * (Vr)^2 = 0.5 * (mass of blue puck) * (velocity of blue puck)^2

From the first equation, we can conclude that the velocity of the red puck after the collision is 0, indicating that the red puck comes to rest after the collision.

Therefore, the magnitude of the velocity of the red puck after the collision is 0.