A two-stage rocket moves in space at a constant velocity of 4650 m/s. The two stages are then separated by a small explosive charge placed between them. Immediately after the explosion the velocity of the 1240-kg upper stage is 5640 m/s in the same direction as before the explosion. What is the velocity (magnitude and direction) of the 2320-kg lower stage after the explosion?

To find the velocity of the lower stage after the explosion, we can use the principle of conservation of momentum. According to this principle, the momentum before and after the explosion should be equal.

The momentum of an object is given by the product of its mass and velocity:

Momentum (P) = Mass (m) × Velocity (v)

Before the explosion, the momentum of the two-stage rocket is the sum of the momenta of the upper and lower stages:

P_initial = P_upper + P_lower

After the explosion, the momentum is still conserved, so we have:

P_final = P_upper + P_lower

Now, let's calculate the initial momentum of the rocket before the explosion:

P_initial = (mass of upper stage) × (velocity of upper stage) + (mass of lower stage) × (velocity of lower stage)

P_initial = (1240 kg) × (4650 m/s) + (2320 kg) × (4650 m/s)

P_initial = 5,754,000 kg·m/s + 10,788,000 kg·m/s

P_initial = 16,542,000 kg·m/s

Since momentum is conserved, the final momentum will be the same:

P_final = 16,542,000 kg·m/s

Now, we can find the velocity of the lower stage after the explosion. Rearranging the momentum equation, we have:

P_final = (mass of upper stage) × (velocity of upper stage) + (mass of lower stage) × (velocity of lower stage)

16,542,000 kg·m/s = (1240 kg) × (5640 m/s) + (2320 kg) × (velocity of lower stage)

Substituting the given values, we can solve for the velocity of the lower stage:

16,542,000 kg·m/s = 6,985,600 kg·m/s + (2320 kg) × (velocity of lower stage)

9,556,400 kg·m/s = (2320 kg) × (velocity of lower stage)

Dividing both sides by 2320 kg, we find:

velocity of lower stage = 9,556,400 kg·m/s / 2320 kg

velocity of lower stage ≈ 4119.3 m/s

Therefore, the velocity of the lower stage after the explosion is approximately 4119.3 m/s in the same direction as before the explosion.