With the engines off, a spaceship is coasting at a velocity of +210 m/s through outer space. It fires a rocket straight ahead at an enemy vessel. The mass of the rocket is 1350 kg, and the mass of the spaceship (not including the rocket) is 2.5 106 kg. The firing of the rocket brings the spaceship to a halt. What is the velocity of the rocket?
The rocket acquires all of the original momentum of the spaceship stops.
(M + m)V1 = m V2
2.5*10^6* 210 = 1350*V2
V2 = 3.89*10^5 m/s
This is an unrealistically high velocity for a rocket, or even a bullet-fired projectile. Furthermore, the spaceship fired a real rocket, most of the velcoty of the rocket would be acquired after release, and would not affect the velocity of the spacecraft. In that case, momentum conservation could not be used.
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the firing of the rocket is equal to the total momentum after the firing.
Given:
Mass of the rocket (m1) = 1350 kg
Mass of the spaceship (m2) = 2.5 * 10^6 kg
Initial velocity of the spaceship (v2i) = +210 m/s
Final velocity of the spaceship (v2f) = 0 m/s (since it comes to a halt)
Final velocity of the rocket (v1f) = ?
Using the conservation of momentum formula:
(m1 * v1f) + (m2 * v2f) = (m1 * v1i) + (m2 * v2i)
Substituting the known values:
(1350 kg * v1f) + (2.5 * 10^6 kg * 0 m/s) = (1350 kg * 0 m/s) + (2.5 * 10^6 kg * 210 m/s)
Simplifying the equation:
1350 kg * v1f = 2.5 * 10^6 kg * 210 m/s
Dividing both sides by 1350 kg:
v1f = (2.5 * 10^6 kg * 210 m/s) / 1350 kg
Calculating the velocity of the rocket:
v1f = 3900000 m/s
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the rocket is fired is equal to the total momentum after the rocket is fired.
Before the rocket is fired, the spaceship's momentum can be calculated by multiplying its mass (2.5 x 10^6 kg) by its velocity (+210 m/s). This gives us a momentum of 2.5 x 10^6 kg x 210 m/s = 5.25 x 10^8 kg·m/s.
After the rocket is fired, the combined momentum of the spaceship and the rocket will be zero since the spaceship comes to a halt. The momentum of the rocket can be calculated by multiplying its mass (1350 kg) by its velocity (v).
So, we have:
Total momentum before = Total momentum after
5.25 x 10^8 kg·m/s = 0 + 1350 kg × v
We can now solve for v by rearranging the equation:
v = (5.25 x 10^8 kg·m/s) / 1350 kg
v ≈ 389,000 m/s
Therefore, the velocity of the rocket after it is fired is approximately +389,000 m/s.