Question

A girl launches a toy rocket from the ground. The engine experiences an average thrust of 5.26 N. The mass of the engine plus fuel before liftoff is 25.0 g, which includes fuel mass of 12.7 g. The engine fires for a total of 1.90 s. (Assume all the fuel is consumed.) (a) Calculate the average exhaust speed of the engine (in m/s). (b) This engine is situated in a rocket case of mass 63.0 g. What is the magnitude of the final velocity of the rocket (in m/s) if it were to be fired from rest in outer space with the same amount of fuel

Answer 1

The average exhaust speed of the engine is calculated to be approximately 786.929 m/s, and the magnitude of the final velocity of the rocket if fired in outer space is roughly 94.916 m/s.

Step-by-step explanation:

To calculate the average exhaust speed (v_ex), we can use the impulse-momentum theorem, which states that impulse is equal to the change in momentum of the system. Impulse is given by the product of the average force exerted by the engine (thrust) and the time interval during which the thrust is applied. If all the fuel is consumed, the change in mass (Δm) is the mass of the fuel.

Impulse = Thrust × Time = 5.26 N × 1.90 s = 9.994 N·s

The momentum change is equal to the mass of the fuel expelled times the average exhaust speed.

Δ(momentum) = Δm × v_ex

Substituting the impulse and solving for v_ex, we get:

v_ex = Impulse / Δm

v_ex = 9.994 N·s / 0.0127 kg = 786.929 m/s

Part B: Final Velocity of the Rocket in Space

The final velocity (V_final) of the rocket in space can be determined using the rocket equation, also known as Tsiolkovsky's rocket equation:

V_final = v_ex × ln(m_initial / m_final)

Where:

• m_initial = mass of the rocket with fuel = 25.0 g + 63.0 g = 88.0 g = 0.088 kg
• m_final = mass of the rocket without fuel = 25.0 g - 12.7 g + 63.0 g = 75.3 g = 0.0753 kg

Calculating the final velocity:

V_final = 786.929 m/s × ln(0.088 kg / 0.0753 kg) ≈ 94.916 m/s

Answer 2

a) v = 786.93 m/s

b) v = 122.40 m/s

Step-by-step explanation:

a) To find the average exhaust speed (v) of the engine we can use the following equation:

`F = (v\Delta m)/(\Delta t)`

Where:

F: is the thrust by the engine = 5.26 N

Δm: is the mass of the fuel = 12.7 g

Δt: is the time of the burning of fuel = 1.90 s

`v = (F*\Delta t)/(\Delta m) = (5.26 N*1.90 s)/(12.7 \cdot 10^(-3) kg) = 786.93 m/s`

b) To calculate the final velocity of the rocket we need to find the acceleration.

The acceleration (a) can be calculated as follows:

`a = (F)/(m)`

In the above equation, m is an average between the mass of the engine plus the rocket case mass and the mass of the engine plus the rocket case minus the fuel mass:

`m = ((m_(engine) + m_(rocket)) + (m_(engine) + m_(rocket) - m_(fuel)))/(2) = (2*m_(engine) + 2*m_(rocket) - m_(fuel))/(2) = (2*25.0 g + 2*63.0 g - 12.7 g)/(2) = 81.65 g`

Now, the acceleration is:

`a = (5.26 N)/(81.65 \cdot 10^(-3) kg) = 64.42 m*s^(-2)`

Finally, the final velocity of the rocket can be calculated using the following kinematic equation:

`v_(f) = v_(0) + at = 0 + 64.42 m*s^(-2)*1.90 s = 122.40 m/s`

I hope it helps you!