A jet aircraft is traveling at 180 m/s in horizontal flight. The engine takes in air at a rate of 77.0 kg/s and burns fuel at a rate of 3.15 kg/s. The exhaust gases are ejected at 595 m/s relative to the speed of the aircraft. Find the thrust of the jet engine. (Take the direction the jet is going to be the positive direction. Indicate the direction with the sign of your answer.)

The correct answer is 33.8 kN can someone show me how to get that answer??

To find the thrust of the jet engine, we can use the conservation of momentum principle. By this principle, the rate of change of momentum of the exhaust gases must equal the total force acting on the system. We have to find the net momentum flux and then the thrust.

First, let's find the momentum flux of the incoming air.
momentum_flux_in = (mass_flow_rate_air) * (velocity_air)
momentum_flux_in = (77.0 kg/s) * (180 m/s) = 13860 kg*m/s²

Then, we'll find the momentum flux of the exhaust gases.
momentum_flux_exhaust = (mass_flow_rate_exhaust) * (velocity_exhaust)
mass_flow_rate_exhaust = (77.0 kg/s + 3.15 kg/s) = 80.15 kg/s
velocity_exhaust = (180 m/s + 595 m/s) = 775 m/s
momentum_flux_exhaust = (80.15 kg/s) * (775 m/s) = 62096.25 kg*m/s²

Now we can find the net momentum flux or change in momentum per unit time (dm/dt).
net_momentum_flux= momentum_flux_exhaust - momentum_flux_in
net_momentum_flux = 62096.25 kg*m/s² - 13860 kg*m/s² = 48236.25 kg*m/s²

Finally, we can calculate the thrust of the jet engine using the definition of thrust (T).
T = (dm/dt)

Plugging the net momentum flux we found above,
T = 48236.25 kg*m/s²

To express the thrust in kiloNewtons (kN), we have to divide it by 1000.
T = 48236.25 kg*m/s² / 1000 = 48.23625 N.

However, the correct answer is given as 33.8 kN. So, somewhere we have made a mistake. Let's verify the solution.

At second glance, we notice that we have made an error in calculating the velocity of exhaust gases. It was given that exhaust gases are ejected at 595 m/s *relative to the speed of the aircraft*.

This means that we should use 595 m/s as the velocity of the exhaust gases instead of adding it to the aircraft's speed. Now let's correct the calculations.

momentum_flux_exhaust = (mass_flow_rate_exhaust) * (velocity_exhaust_relative)
momentum_flux_exhaust = (80.15 kg/s) * (595 m/s) = 47689.25 kg*m/s²

Now we can find the net momentum flux with the corrected values.
net_momentum_flux = momentum_flux_exhaust - momentum_flux_in
net_momentum_flux = 47689.25 kg*m/s² - 13860 kg*m/s² = 33829.25 kg*m/s²

Now, we can calculate the thrust of the jet engine.
T = (dm/dt) = 33829.25 kg*m/s²

Converting the thrust to kiloNewtons (kN),
T = 33829.25 kg*m/s² / 1000 = 33.82925 kN

Rounding off to one decimal place, the thrust of the jet engine is 33.8 kN, which matches the correct answer.

To find the thrust of the jet engine, we can use the principle of conservation of momentum. The thrust of the engine is equal to the rate of change of momentum of the exhaust gases.

The momentum of an object is given by the product of its mass and velocity. In this case, the mass of the exhaust gases being ejected per second is given by the fuel consumption rate. So, the mass of the exhaust gases is 3.15 kg/s.

The velocity of the exhaust gases relative to the aircraft is given as 595 m/s.

Using the principle of conservation of momentum, we can find the thrust of the jet engine:

Thrust = Change in momentum / Time

The change in momentum is given by the product of the mass and the change in velocity. In this case, the change in velocity is the velocity of the exhaust gases (595 m/s) minus the velocity of the aircraft (180 m/s).

Change in momentum = mass x (velocity of exhaust gases - velocity of aircraft)
= 3.15 kg/s x (595 m/s - 180 m/s)

Substituting the given values:

Change in momentum = 3.15 kg/s x (595 m/s - 180 m/s)
= 3.15 kg/s x 415 m/s
= 1307.25 kg*m/s

Finally, dividing the change in momentum by time gives the thrust:

Thrust = 1307.25 kg*m/s / 1 s
= 1307.25 N

Converting the thrust to kilonewtons by dividing by 1000:

Thrust = 1307.25 N / 1000
= 1.30725 kN

Therefore, the thrust of the jet engine is approximately 1.30725 kN.

To find the thrust of the jet engine, we can use the principle of conservation of momentum.

The momentum change experienced by the jet engine can be calculated using the formula:

Change in momentum = Final momentum - Initial momentum

Since the jet is in horizontal flight, there is no vertical movement. Therefore, we can assume that the jet's initial and final vertical momentum is zero.

Initial momentum = Final momentum

Now, let's find the initial and final horizontal momenta of the exhaust gases:

Initial horizontal momentum = Mass of the exhaust gases (me) x Velocity relative to the aircraft (ue)

Final horizontal momentum = Mass of the exhaust gases (me) x Final velocity relative to the aircraft (ve)

The change in momentum can be calculated using:

Change in momentum = Final momentum - Initial momentum
= me * ve - me * ue

According to the principle of conservation of momentum, the change in momentum experienced by the exhaust gases is equal and opposite to the momentum change experienced by the jet engine. Therefore:

Change in momentum = Thrust of the jet engine (T) x time (t)

Now, let's substitute the given values into the equation:

me = 77.0 kg/s (mass flow rate of the exhaust gases)
ve = 595 m/s (final velocity of the exhaust gases relative to the aircraft)
ue = 0 m/s (initial velocity of the exhaust gases relative to the aircraft)
T = thrust of the jet engine (to be determined)
t = time (we can assume a value of 1 second)

Change in momentum = T * t

me * ve - me * ue = T * t

(77.0 kg/s * 595 m/s) - (77.0 kg/s * 0 m/s) = T * 1 s

(77.0 kg/s * 595 m/s) = T * 1 s

T = (77.0 kg/s * 595 m/s) / 1 s

T = 45,815 N

Therefore, the thrust of the jet engine is 45.8 kN (since 1 kN = 1000 N).

Note: The given answer of 33.8 kN may be incorrect or there might be additional factors involved not mentioned in the question. However, the calculations above yield a thrust value of 45.8 kN based on the provided information.