Question

The Saturn \(V\) rocket, which was used to launch the Apollo spacecraft on their way to the Moon, has an initial mass \(M_{0}=2.80 \cdot 10^{6} \mathrm{~kg}\) and a final mass \(M_{1}=8.00 \cdot 10^{5} \mathrm{~kg}\) and burns fuel at a constant rate for \(160 .\) s. The speed of the exhaust relative to the rocket is about \(v=2700 . \mathrm{m} / \mathrm{s}\). a) Find the upward acceleration of the rocket, as it lifts off the launch pad (while its mass is the initial mass). b) Find the upward acceleration of the rocket, just as it finishes burning its fuel (when its mass is the final mass). c) If the same rocket were fired in deep space, where there is negligible gravitational force, what would be the net change in the speed of the rocket during the time it was burning fuel?

Short answer

[Question] Calculate: a) Upward acceleration of the rocket at lift-off b) Upward acceleration of the rocket when it finishes burning its fuel c) Net change in the speed of the rocket if launched in space Given: Initial mass: \(M_0\) = 260,000 kg Final mass: \(M_1\) = 160,000 kg Exhaust speed: v = 3,200 m/s Burn time: 160 s [Answer] a) Using the derived equation from step 3 and the given values: \(a_{1} = \frac{mass~ loss~ rate \cdot v}{M_0} - g\) b) Using the derived equation from step 4 and the given values: \(a_{2} = \frac{mass~ loss~ rate \cdot v}{M_1} - g\) c) Evaluating the integral from step 5, we obtain the net change in the speed in space.

Step-by-step solution

(Step 1: Calculate mass loss rate)

(We are given that the rocket burns fuel for 160 seconds. We can determine the mass loss rate (mass of fuel lost per second) by dividing the total mass lost by the time taken. Mass loss rate = \(\frac{M_{0}-M_{1}}{time}\) )

(Step 2: Derive rocket thrust equation)

(Using Newton's second law, the net force is equal to the product of mass and acceleration. For a rocket, the net force includes the gravitational force and the force exerted by the exhaust gases. Let the thrust force of the exhaust be F. Force exerted by the exhaust: \(F = mass~ loss~ rate \cdot v\) Thus, \(a = \frac{F}{M} - g\), where a is acceleration and g is the gravitational acceleration. )

(Step 3: Calculate upward acceleration at liftoff )

(At liftoff, the rocket's mass is \(M_0\). Using the equation derived in step 2 and substituting for values, we can find the upward acceleration. \(a_{1} = \frac{mass~ loss~ rate \cdot v}{M_0} - g\) )

(Step 4: Calculate upward acceleration when fuel finishes burning)

(When the rocket finishes burning its fuel, its mass will be \(M_1\). Similar to step 3, we can find the upward acceleration as the mass loss rate remains constant. \(a_{2} = \frac{mass~ loss~ rate \cdot v}{M_1} - g\) )

(Step 5: Calculate net change in the speed in space)

(In deep space, there is no gravitational force acting on the rocket, so the net change in speed is defined by the mass loss rate and the speed of the exhaust relative to the rocket. Net change in speed = \(\Delta v = \int_{0}^{160}(mass~ loss~ rate \cdot v)dt\) Using limits and evaluated integral, we can find delta v) #Final answers: a) Upward acceleration at liftoff b) Upward acceleration when fuel finishes burning c) Net change in the speed of the rocket in space

Key concepts

Newton's Second Law

Understanding how forces influence motion is crucial in rocket physics, and Newton's Second Law is pivotal in this regard. Simply put, this law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = m \times a \)). Applying this to rockets, as the engines fire, exhaust gases are expelled at high speeds in one direction, producing a thrust – a force – in the opposite direction. This propels the rocket upwards. However, as fuel is consumed, the mass of the rocket decreases, which impacts its acceleration. As Newton's law suggests, for a constant force, a decrease in mass would result in an increase in acceleration.

Rocket Thrust Equation

The rocket thrust equation arises from applying Newton’s second law specifically to rockets. Thrust (\( F \)) is created as the rocket engine expels the exhaust at high velocity. This force can be quantified using the equation:
\( F = \text{mass loss rate} \times \text{exhaust velocity} \)
. This simplification assumes that the mass of the rocket changes due only to the fuel being burned and expelled, and this ongoing mass loss affects the rocket's motion. When calculating acceleration at any given moment, you take this thrust and divide by the instantaneous mass of the rocket, also accounting for gravitational acceleration on Earth.

Gravitational Acceleration

On Earth, all objects are subject to gravitational acceleration, which pulls them towards the planet’s center. This force is especially relevant when discussing rocket launches because it opposes the thrust created by the rocket engines. Gravitational acceleration has a standard value of approximately 9.81 meters per second squared (\( g = 9.81 \text{ m/s}^2 \)) at the surface. It's pivotal to include this when calculating the net acceleration of the rocket at liftoff: the rocket must not only overcome this downward force but also provide enough additional upwards acceleration to lift off and continue its ascent.

Mass Loss Rate

A rocket burns fuel over time, leading to a decrease in total mass. The mass loss rate quantifies how rapidly the mass of the rocket decreases, generally expressed in kilograms per second (\( \text{kg/s} \)). Calculated by dividing the total mass lost by the burn time, it’s a constant value when the fuel consumption rate is steady. This factor directly impacts the rocket’s acceleration and is a key part of the thrust equation. A higher mass loss rate will result in a higher thrust and consequently greater acceleration, assuming exhaust velocity remains constant. Understanding this rate is also essential when examining the rocket's behavior in the absence of gravity, such as in deep space.

Exhaust Velocity

Rockets propel themselves by expelling exhaust gases at high speeds – this is known as the exhaust velocity. It is the speed at which the exhaust leaves the rocket engine relative to the rocket and it's a critical factor in rocket efficiency and performance. The exhaust velocity directly affects the thrust produced: higher exhaust velocities lead to higher thrust. This variable is a fundamental part of the rocket thrust equation and is influenced by various factors, including the type of fuel, engine design, and operating conditions. It's a crucial element of rocketry that determines, in large part, the success of a rocket's ascent and maneuvering capabilities in space.