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 6 % of its mass in 310 s (assume the burn rate is constant).
(a) What is the speed v of the rocket after a burn time of 155.0 s? (suppose that the rocket starts at rest; and enter your answer in m/s)
v=
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(b) What is the instantaneous acceleration a of the rocket at time 155.0 s after the start of the engines?(in m/s2)
a=
To answer these questions, we need to apply the principle of conservation of momentum and use the concept of rocket propulsion.
(a) To find the speed of the rocket after a burn time of 155.0 s, we can consider the change in momentum of the rocket. The initial momentum of the rocket is zero since it starts at rest.
According to the principle of conservation of momentum, the change in momentum of the rocket will be equal to the momentum of the ejected fuel. We know the speed of the ejected fuel with respect to the rocket, vex, which is 1.5 km/s. However, we need to convert it to m/s since we are working in SI units.
1 km = 1000 m, so 1.5 km/s = 1500 m/s.
Now, let's calculate the mass of the fuel ejected during the burn time of 155.0 s. We are given that the rocket burns 6% of its mass in 310 s. Therefore, in 155.0 s, it will burn half of that amount.
To find the mass of the fuel burned, we need to know the initial mass of the rocket. Let's assume it is m0 kg.
The mass of the fuel burned in 310 s is (6/100) * m0 = (0.06 * m0) kg.
Therefore, the mass of the fuel burned in 155.0 s is (0.06 * m0) / 2 = (0.03 * m0) kg.
Now we can calculate the change in momentum:
Change in momentum = (mass of the fuel burned) * (velocity of the ejected fuel)
Change in momentum = (0.03 * m0) kg * (1500 m/s)
Since momentum is equal to mass times velocity, the change in momentum is equivalent to the change in mass of the rocket times its final velocity.
Change in momentum = Δm * vf.
The change in mass of the rocket is equal to the mass of the fuel burned, which is (0.03 * m0) kg. The final velocity of the rocket is what we need to find, so let's denote it as v.
Therefore, we can write the equation as follows:
(0.03 * m0) kg * (1500 m/s) = (0.03 * m0) kg * v.
Simplifying the equation, we find:
1500 m/s = v.
So the speed of the rocket after a burn time of 155.0 s is 1500 m/s.
(b) To find the instantaneous acceleration of the rocket at time 155.0 s, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass times its acceleration.
The force causing the acceleration of the rocket is provided by the burning of the fuel. It is given by:
Force = (burn rate) * (mass of the fuel burned) * (ejection velocity of the fuel).
In this case, the burn rate is constant and equal to 0.06, and the mass of the fuel burned is (0.03 * m0) kg.
Therefore, the force is equal to:
Force = 0.06 * (0.03 * m0) kg * 1500 m/s.
Now, we can use Newton's second law to find the acceleration:
Force = mass * acceleration.
Substituting the expressions for force and mass, we get:
0.06 * (0.03 * m0) kg * 1500 m/s = m0 kg * acceleration.
Simplifying the equation, we find:
0.06 * 0.03 * 1500 m/s = acceleration.
So the instantaneous acceleration of the rocket at time 155.0 s is 27 m/s².