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The space probe Deep Space 1 was launched on October 24,1998 . Its mass was \(474 \mathrm{kg}\). The goal of the mission was to test a new kind o engine called an ion propulsion drive. This engine generated only a weal thrust, but it could do so over long periods of time with the consumptio of only small amounts of fuel. The mission was spectacularly successful At a thrust of \(56 \mathrm{mN}\) how many days were required for the probe to attair a velocity of \(805 \mathrm{m} / \mathrm{s}(1800 \mathrm{mi} / \mathrm{h}),\) assuming that the probe started fron rest and that the mass remained nearly constant?

Short Answer

Expert verified
Approximately 79 days are required.

Step by step solution

01

Identify Given Values

The mass of the probe is given as \(474 \text{ kg}\), the thrust is \(56 \text{ mN}\), and the desired velocity is \(805 \text{ m/s}\). We're asked to calculate the time required to reach this velocity. The probe is assumed to start from rest.
02

Convert Thrust to Newtons

The thrust is given in millinewtons (\(56 \text{ mN}\)). We convert this to newtons: \[56 \text{ mN} = 56 \times 10^{-3} \text{ N} = 0.056 \text{ N}.\]
03

Calculate Acceleration

We use Newton's second law \(F = ma\) to find the acceleration (\(a\)). Rearranging the formula gives us \(a = \frac{F}{m}\). Substituting the known values: \[a = \frac{0.056 \text{ N}}{474 \text{ kg}} \approx 1.181 \times 10^{-4} \text{ m/s}^2.\]
04

Determine Time Using Kinematics

Using the kinematic equation \(v = at\), where \(v\) is the velocity (805 m/s), \(a\) is the acceleration, and \(t\) is the time, we solve for \(t\): \[t = \frac{v}{a} = \frac{805 \text{ m/s}}{1.181 \times 10^{-4} \text{ m/s}^2} \approx 6,818,053.34 \text{ seconds}.\]
05

Convert Time from Seconds to Days

To convert seconds to days, use the conversion \(1 \text{ day} = 86,400 \text{ seconds}\). \[t = \frac{6,818,053.34}{86,400} \approx 78.9 \text{ days}.\]

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Key Concepts

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

Space Probe
A space probe is a robotic spacecraft that leaves Earth's atmosphere to gather scientific data. Unlike satellites, which orbit Earth, space probes venture into deeper space.
The mission of Deep Space 1 was groundbreaking, as it was not only meant to explore but to test innovative propulsion technology.
Ion propulsion systems were particularly exciting because they could propel a probe using small amounts of fuel over extended periods.
  • Space probes are essential for exploring moons, planets, and other celestial bodies in our solar system.
  • They perform tasks such as imaging, taking samples, and measuring environmental conditions.
Thanks to its ion propulsion system, Deep Space 1 set a precedent for future missions that rely on efficient, long-duration travel.
Kinematics
Kinematics is the branch of physics concerned with describing motion using the language of displacement, velocity, and acceleration. It's especially relevant in calculating how objects like space probes move.
In the case of Deep Space 1, the probe starts from rest, meaning its initial velocity is zero. With the help of kinematic equations, one can determine how long it takes to reach a certain speed given a constant acceleration.
The process involves using the formula \[ v = at \] where \(v\) is velocity, \(a\) is acceleration, and \(t\) is time. Given these parameters, you can solve for any one of these three values if the other two are known.
  • Initial velocity is vital when setting the motion equations.
  • Each kinematic variable affects how the probe behaves in space.
This is critical for plotting the journey of probes like Deep Space 1 to ensure they meet their mission timelines.
Newton's Second Law
Newton's second law of motion is fundamental in understanding how forces affect an object's motion. Mathematically, it is represented as: \[ F = ma \] where \(F\) is the force applied to the object, \(m\) is its mass, and \(a\) is the acceleration produced.
For space missions like Deep Space 1, understanding this law is crucial in determining how much thrust—a type of force—is necessary to achieve the desired motion.
The thrust of Deep Space 1's ion propulsion engine was \(0.056\) Newtons, a small force but effective over long periods due to the low fuel requirement.
  • By rearranging the formula to \( a = \frac{F}{m} \), we calculate the acceleration.
  • This calculation provides essential insights into how the spacecraft's speed will change over time.
  • This law helps engineers design engines capable of sustaining prolonged missions in space.
Comprehensive understanding of this law ensures that space probes can successfully navigate the vast reaches of space.

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Most popular questions from this chapter

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