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=

(b) What is the instantaneous acceleration a of the rocket at time 140.0 s after the start of the engines?(in m/s2)

a=

To solve this problem, we can use the principle of conservation of momentum. The momentum of the rocket and the burned fuel before and after the burn time must be equal.

Let's first find the velocity of the rocket after 140.0 s.

(a) Speed of the rocket after a burn time of 140.0 s:
1. Convert the ejected fuel velocity from km/s to m/s:
v_ex = 1.5 km/s = 1500 m/s.
2. Calculate the mass of the remaining rocket after burning:
mass_remaining = 100% - burn rate = 100% - 8% = 92%
mass_remaining = 92% of original mass of the rocket.
3. Calculate the mass ejected in 140.0 s:
mass_ejected = burn rate × original mass of the rocket × (burn time / total time)
mass_ejected = 0.08 × original mass of the rocket × (140.0 s / 280.0 s)
4. Calculate the final momentum of the system:
momentum_before = initial mass of the system × initial velocity of the system
momentum_after = (mass_remaining + mass_ejected) × velocity of the rocket after burn time + mass_ejected × velocity of the fuel
5. Set the momenta equal to each other and solve for the velocity of the rocket after 140.0 s.

v
Now let's calculate the acceleration of the rocket at time 140.0 s:

(b) Instantaneous acceleration of the rocket at time 140.0 s:
1. Derive the equation for velocity with respect to time by solving for v in part (a).
2. Differentiate the velocity equation with respect to time to find the acceleration equation.
3. Substitute the time (140.0 s) into the acceleration equation to find the instantaneous acceleration at that time.

a

By following these steps, you should be able to find the speed and acceleration of the rocket after a burn time of 140.0 s.