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In his 1865 science fiction novel From the Earth to the Moon, Jules Verne described how three astronauts are shot to the Moon by means of a huge gun. According to Verne, the aluminum capsule containing the astronauts is accelerated by ignition of nitrocellulose to a speed of \(11 \mathrm{~km} / \mathrm{s}\) along the gun barrel's length of \(220 \mathrm{~m} .\) (a) In \(g\) units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts? A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston along the gun's tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves \(3.5 \mathrm{~km}\) and reaches a speed of \(7.0 \mathrm{~km} / \mathrm{s}\). Once launched, the rocket can be fired to gain additional speed. (c) In \(g\) units, what would be the average acceleration of the rocket within the launcher? (d) How much additional speed is needed (via the rocket engine) if the rocket is to orbit Earth at an altitude of \(700 \mathrm{~km} ?\)

Step by step solution

01

Identify Given Values (Part a)

We are asked to find the average acceleration in \( g \) units for the capsule in Verne's scenario. The capsule's final speed \( v = 11 \text{ km/s} = 11000 \text{ m/s} \), and the gun barrel's length \( s = 220 \text{ m} \). The initial speed \( u = 0 \text{ m/s} \).

02

Calculate Average Acceleration (Part a)

We use the formula for acceleration \( a \) with constant acceleration: \ \( v^2 = u^2 + 2as \). Substitute the values to solve for \( a \). \ \( (11000)^2 = 0 + 2a\times220 \). \ \( 121000000 = 440a \). \ \( a = \frac{121000000}{440} \approx 275000 \text{ m/s}^2 \).

03

Convert to g Units (Part a)

1 \( g \) equals \( 9.8 \text{ m/s}^2 \). To find the acceleration in \( g \) units, divide by \( 9.8 \). \ \( a_{g} = \frac{275000}{9.8} \approx 28061 \, g \).

04

Analyze Tolerability (Part b)

For humans, an acceleration of above around \( 10g \) can be lethal, making \( 28061g \) certainly intolerable.

05

Identify Given Values (Part c)

For the SHARP gun, the distance \( s = 3.5 \text{ km} = 3500 \text{ m} \) and final velocity \( v = 7.0 \text{ km/s} = 7000 \text{ m/s} \). Initial velocity \( u = 0 \text{ m/s} \).

06

Calculate Average Acceleration (Part c)

Using the same formula: \ \( v^2 = u^2 + 2as \), substitute the values. \ \( (7000)^2 = 0 + 2a\times3500 \). \ \( 49000000 = 7000a \). \ \( a = \frac{49000000}{7000} \approx 7000 \text{ m/s}^2 \).

07

Convert to g Units (Part c)

Convert the acceleration to \( g \) units: \ \( a_{g} = \frac{7000}{9.8} \approx 714.3 \, g \).

08

Identify Required Speed for Orbit (Part d)

To find the additional speed needed for orbit at 700 km altitude, calculate the orbital speed.

09

Calculate Orbital Speed (Part d)

Use the formula for orbital speed: \[ v = \sqrt{\frac{GM}{R}} \] \ where \( G = 6.674\times10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \), \( M = 5.972\times10^{24} \text{ kg} \) (Earth's mass), and \( R = 6771 \text{ km} = 6771\times10^3 \text{ m} \) (Earth's radius plus 700 km). \ \( v_{orbital} \approx 7500 \text{ m/s} \).

10

Compute Additional Speed (Part d)

The launcher achieves \( 7000 \text{ m/s} \), so the extra speed required is \ \( v_{extra} = 7500 - 7000 = 500 \text{ m/s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acceleration Calculation

Acceleration is a key concept in physics, denoting how quickly an object鈥檚 velocity changes. When solving for acceleration with constant motion, we use the formula: \[ v^2 = u^2 + 2as \] where:

• \(v\) is the final velocity,
• \(u\) is the initial velocity,
• \(a\) is the acceleration,
• and \(s\) is the distance covered.

In Verne's scenario, the capsule's final speed is \(11,000\) m/s, and the barrel鈥檚 length is \(220\) m. Initial speed \(u\) is \(0\) m/s, as it starts from rest. Plugging these into the equation gives: \[ (11,000)^2 = 0 + 2a \times 220 \] This simplifies to yield an acceleration \(a\) of approximately \(275,000 \text{ m/s}^2\). This high number reflects the immense speed change over the short barrel distance.

Conversion to g units

In physics, acceleration is often expressed in terms of \(g\) units for ease of comparison to the gravitational acceleration at Earth's surface (\(1g\) is \(9.8\) m/s虏). To convert an acceleration value into \(g\) units, divide it by \(9.8\). For the capsule, with an acceleration of \(275,000 \text{ m/s}^2\), the conversion process looks like this: \[ a_{g} = \frac{275,000}{9.8} \approx 28,061\, g \] This indicates an enormous acceleration, far beyond tolerable human limits, suggesting intense forces acting on the capsule.

Orbital Speed Calculation

To place an object in orbit around Earth, it must reach an orbital speed which depends on the altitude of the orbit. This is calculated using the formula: \[ v = \sqrt{\frac{GM}{R}} \] where:

• \(G = 6.674\times10^{-11}\) m鲁kg鈦宦箂鈦宦� (universal gravitational constant),
• \(M = 5.972\times10^{24}\) kg (Earth's mass),
• \(R = 6,371 + 700 = 7,071\) km or \(7,071\times10^3\) m (Earth's radius plus 700 km altitude).

Substituting into the equation, we solve for \(v\): \[ v \approx 7,500 \text{ m/s} \] In this context, if the rocket reaches \(7,000 \text{ m/s}\) from its initial launch, an additional \(500 \text{ m/s}\) is needed to achieve orbital velocity. Efficient rocket stage ignition after launch can provide this required boost.