Model rocket engines are sized by thrust, thrust duration, and total impulse, among other characteristics. A size C5 model rocket engine has an average thrust of 5.26 N, a fuel mass of 12.7 g, and an initial mass of 25.5 g. The duration of its burn is 1.90 s.
(a) What is the average exhaust speed of the engine?
(b) If this engine is placed in a rocket body of mass 57.5 g, and if the rocket is fired in outer space, what is the final velocity of the rocket? Assume the fuel burns at a constant rate
for average speed i did fthrust=change of M/change in T i got .78 this is not right help please
What units are you using.
Mass change is .0127kg,
force =5.26N
deltTime=1.90sec.
v=F(time)/masschange
To find the average exhaust speed of the engine, you can use the rocket equation:
v = (u - (m_dot * v_e)) / (m - m_dot)
where:
v = average exhaust speed
u = initial velocity of the rocket
m_dot = mass flow rate of the engine (fuel mass / burn time)
v_e = exhaust speed of the engine
m = initial mass of the rocket (including the fuel mass)
Let's calculate it step-by-step:
(a) Average Exhaust Speed:
First, calculate the mass flow rate of the engine (m_dot):
m_dot = 12.7 g / 1.90 s
m_dot = 6.68 g/s
Next, substitute the values into the rocket equation:
v = (u - (m_dot * v_e)) / (m - m_dot)
Since the rocket starts at rest in this case (u = 0), the equation simplifies to:
v = -(m_dot * v_e) / (m - m_dot)
The initial mass of the rocket (m) can be calculated by adding the fuel mass and the rocket body mass:
m = 25.5 g + 57.5 g
m = 83 g
Now, substitute the known values:
v = -((6.68 g/s) * v_e) / (83 g - 6.68 g/s)
Simplifying further:
v = -6.68 v_e / (76.32 g/s)
Given that the average thrust (F) of the engine is equal to mass flow rate (m_dot) multiplied by the exhaust speed (v_e):
F = m_dot * v_e
5.26 N = (6.68 g/s) * v_e
Thus, the average exhaust speed can be found by rearranging the equation:
v_e = 5.26 N / (6.68 g/s)
v_e = 0.787 Ns/g
So, the average exhaust speed of the engine is approximately 0.787 Ns/g.
(b) Final Velocity of the Rocket:
To find the final velocity of the rocket, we need to use the law of conservation of momentum.
The initial momentum of the system (rocket + exhaust gas) is zero because the rocket starts at rest:
P_initial = m * u + (m_dot * v_e) * t = 0
The final momentum of the system is given by:
P_final = (m + m_dot * t) * v
Since the initial momentum is zero: P_initial = 0
Substituting the known values:
0 = (83 g + (6.68 g/s) * 1.90 s) * v
Simplifying further:
0 = (83 g + 12.692 g) * v
0 = 95.692 g * v
Now, divide both sides by 95.692 g:
v = 0 m/s
Therefore, the final velocity of the rocket in outer space would be 0 m/s.
To find the average exhaust speed of the engine, we can use the concept of conservation of momentum.
The formula for average exhaust speed (Ve) is given by:
Ve = (2 * Thrust) / (Fuel Mass)
Using the given values, we have:
Thrust = 5.26 N
Fuel Mass = 12.7 g
First, we need to convert the fuel mass from grams to kilograms:
Fuel Mass = 12.7 g = 0.0127 kg
Plugging in the values into the formula for Ve, we get:
Ve = (2 * 5.26 N) / 0.0127 kg
Ve = 827.56 N/kg
Therefore, the average exhaust speed of the engine is 827.56 m/s.
Now, let's move on to finding the final velocity of the rocket. We can use the concept of the rocket equation, which states:
V = Ve * ln(Mi / Mf)
where:
V is the velocity of the rocket
Ve is the average exhaust speed of the engine (calculated previously)
Mi is the initial mass of the rocket (rocket body mass + fuel mass)
Mf is the final mass of the rocket (rocket body mass after the fuel is burnt)
Given values:
Rocket body mass = 57.5 g = 0.0575 kg (convert to kilograms)
First, we need to calculate the initial mass of the rocket:
Mi = Rocket body mass + Fuel mass
Mi = 0.0575 kg + 0.0127 kg
Mi = 0.0702 kg
Next, we need to calculate the final mass of the rocket after the fuel is burnt:
Mf = Rocket body mass
Mf = 0.0575 kg
Now, plug in the values into the rocket equation:
V = 827.56 m/s * ln(0.0702 kg / 0.0575 kg)
Calculating the natural logarithm:
ln(0.0702 kg / 0.0575 kg) ≈ 0.1853
V ≈ 827.56 m/s * 0.1853
V ≈ 153.12 m/s
Therefore, the final velocity of the rocket, when fired in outer space, is approximately 153.12 m/s.