Consider a rocket in space that ejects burned fuel at a speed of vex= 1.5 km/s with respect to the rocket. The rocket burns 8 % of its mass in 280 s (assume the burn rate is constant).
(a) What is the speed v of the rocket after a burn time of 140.0 s? (suppose that the rocket starts at rest; and enter your answer in m/s)
v=
Consider a rocket in space that ejects burned fuel at a speed of vex= 1.5 km/s with respect to the rocket. The rocket burns 8 % of its mass in 280 s (assume the burn rate is constant).
(a) What is the speed v of the rocket after a burn time of 140.0 s? (suppose that the rocket starts at rest; and enter your answer in m/s)
v=
To find the speed v of the rocket after a burn time of 140.0 s, we can use the principle of conservation of momentum.
The momentum of an object is equal to its mass multiplied by its velocity: momentum = mass × velocity. In this case, we need to consider both the mass of the rocket and the mass of the burned fuel.
Let's first calculate the mass of the fuel burned during the 140.0 s. Given that the rocket burns 8% of its mass in 280 s, we can find the mass burned in 140 s by using the proportion:
(8/100) × mass of the rocket = x mass burned in 280 s
(8/100) × mass of the rocket = (x/140) × mass burned in 140 s
Cross multiplying, we get:
140 × (8/100) × mass of the rocket = 280 × x
11.2 × mass of the rocket = 280 × x
Dividing both sides by 11.2, we find:
mass of the rocket = (280 × x) / 11.2
Now, let's use the principle of conservation of momentum. At the beginning, the rocket is at rest, so its initial momentum is zero. After the burn time of 140.0 s, the rocket has ejected fuel with a velocity of vex = 1.5 km/s relative to the rocket.
The change in momentum of the rocket is equal to the momentum lost by the fuel. Therefore, we have:
change in momentum of the rocket = momentum lost by fuel
mass of the rocket × final velocity of the rocket = mass burned × velocity of the burned fuel
Substituting the values we have:
mass of the rocket × v = (280 × x) / 11.2 × 1.5 km/s
To convert km/s to m/s, we multiply by 1000:
mass of the rocket × v = (280 × x) / 11.2 × 1.5 km/s × 1000 m/s
mass of the rocket × v = (280 × x) / 11.2 × 1500 m/s
mass of the rocket × v = (280 × x) / 16.8 m/s
Now, we know that the mass burned in 140 s is given by (280 × x) / 11.2. So we substitute that in:
mass of the rocket × v = (280 × (280 × x) / 11.2) / 16.8 m/s
Simplifying:
mass of the rocket × v = (280 × 280 × x) / (11.2 × 16.8) m/s
Now, we have the equation:
mass of the rocket × v = (280 × 280 × x) / (11.2 × 16.8) m/s
To solve for v, we need to know the mass of the rocket. Once we have that, we can plug in the values and calculate the speed v of the rocket after a burn time of 140.0 s.