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184_notes:examples:week3_spaceship_asteroid [2018/02/03 18:44] – tallpaul | 184_notes:examples:week3_spaceship_asteroid [2018/05/24 15:07] – curdemma | ||
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- | =====Preventing an Asteroid Collision===== | + | [[184_notes: |
+ | |||
+ | =====Example: | ||
Suppose your friend is vacationing in Italy, and she has lent you her spaceship for the weekend. You have gathered together a group of friends and you are currently cruising through the heavens together and having a great time. You are surrounded by nothingness in all directions. Suddenly, the radar starts beeping ferociously. The ship is on a collision course with an asteroid. You are not too worried about survival -- the ship is practically indestructible. However, you know your friend would be devastated if you returned her spaceship with a scratch or dent from the asteroid. You need to prevent the collision. | Suppose your friend is vacationing in Italy, and she has lent you her spaceship for the weekend. You have gathered together a group of friends and you are currently cruising through the heavens together and having a great time. You are surrounded by nothingness in all directions. Suddenly, the radar starts beeping ferociously. The ship is on a collision course with an asteroid. You are not too worried about survival -- the ship is practically indestructible. However, you know your friend would be devastated if you returned her spaceship with a scratch or dent from the asteroid. You need to prevent the collision. | ||
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===Goal=== | ===Goal=== | ||
* Prevent the asteroid collision using the long-distance wiring setup. | * Prevent the asteroid collision using the long-distance wiring setup. | ||
- | |||
- | ===f=== | ||
- | * The current distance between the ship and the asteroid. | ||
- | * The distribution of charge on the asteroid. | ||
- | * The distribution of charge on the central component and on the ship itself. | ||
===Representations=== | ===Representations=== | ||
{{ 184_notes: | {{ 184_notes: | ||
- | + | <WRAP TIP> | |
- | {{ 184_notes: | + | === Approximations |
- | + | We want to make a useful representation, | |
- | ====Solution==== | + | |
- | + | ||
- | ===Approximations & Assumptions=== | + | |
* We approximate the asteroid as a point charge. | * We approximate the asteroid as a point charge. | ||
* We approximate the ship as a rectangle (as seen in the representation below). | * We approximate the ship as a rectangle (as seen in the representation below). | ||
* We approximate the central component as a point charge. | * We approximate the central component as a point charge. | ||
- | * We assume the long-distance wiring setup is perfectly efficient. That is, no charge is lost to space and the charge of the asteroid and the charge of the central component will always add to $50 \text{ C}$. | ||
- | * We assume that the rest of the ship is neutral. | ||
* We approximate that the path of the asteroid goes straight toward the central component. | * We approximate that the path of the asteroid goes straight toward the central component. | ||
- | * We assume | + | These are all reasonable approximations, |
+ | </ | ||
+ | {{ 184_notes: | ||
+ | |||
+ | ====Solution==== | ||
+ | This is a complicated problem. We will definitely need a plan. Before we dive into the numbers, let's describe what will happen qualitatively. If we can extract some charge from the asteroid, then both the asteroid and the central component will be positively charged. Two positive point charges will repel, so as the asteroid gets closer to the ship, it slows down, until eventually (hopefully!) it will come to stop, and start moving away from the ship. The thing we are most concerned about is how close the asteroid will approach before turning around -- we don't want it to get close enough to scrape the side of the ship. It might make sense to take a conservation-of-energy | ||
+ | |||
+ | <WRAP TIP> | ||
+ | === Plan === | ||
+ | We will use conservation | ||
+ | * The system is the asteroid. | ||
+ | * The initial state is when the asteroid is very far away, an hour from impact. | ||
+ | * The final state is when the asteroid has stopped before crashing into the ship. | ||
+ | * We expect the system to experience a increase in electric potential energy, | ||
+ | * We can use the change in kinetic energy to find the change in electric potential energy, which can be used to find the charge needed on the central component. | ||
+ | </ | ||
+ | |||
+ | The asteroid' | ||
+ | |||
+ | <WRAP TIP> | ||
+ | === Assumptions === | ||
+ | We did not include the ship in our system. What if its energy changes due to the incoming asteroid? Well, in order to simplify problem, we will just assume its kinetic energy doesn' | ||
* The ship is currently floating through space, and therefore has constant velocity. | * The ship is currently floating through space, and therefore has constant velocity. | ||
* The ship is far more massive than the asteroid to the degree that its current constant-velocity motion is not affected by the asteroid. | * The ship is far more massive than the asteroid to the degree that its current constant-velocity motion is not affected by the asteroid. | ||
+ | Based on our representations, | ||
+ | </ | ||
- | We choose to solve this example using energy. The system is the asteroid and ship with nothing in the surroundings, | + | The change in electric potential energy will depend on how close the asteroid gets to the ship, and how we choose to charge the central component. Currently, its distance is $4000 \text{ m/s}\cdot 60 \text{ |
- | + | ||
- | The change in electric potential energy will depend on how close the asteroid gets to the ship, and how we choose to charge the central component. Currently, its distance is $4000 \text{ m/s}\cdot 60 \text{ | + | |
\begin{align*} | \begin{align*} | ||
\Delta U &= \frac{1}{4\pi\epsilon_0}\frac{q_{ast}q_{comp}}{x_f} - \frac{1}{4\pi\epsilon_0}\frac{q_{ast}q_{comp}}{x_i} \\ | \Delta U &= \frac{1}{4\pi\epsilon_0}\frac{q_{ast}q_{comp}}{x_f} - \frac{1}{4\pi\epsilon_0}\frac{q_{ast}q_{comp}}{x_i} \\ | ||
- | & | + | & |
\end{align*} | \end{align*} | ||
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$$q_{comp}(50 \text{ C}-q_{comp}) = 50 \text{ C} \cdot q_{comp} - q_{comp}^2 > 530 \text{ C}^2$$ | $$q_{comp}(50 \text{ C}-q_{comp}) = 50 \text{ C} \cdot q_{comp} - q_{comp}^2 > 530 \text{ C}^2$$ | ||
- | A simple guess of $q_{comp}=q_{ast}=25\text{ C}$ yields $q_{comp}q_{ast} = 625 \text{ C}^2 > 530 \text{ C}^2$, which is enough to save the ship from cosmetic damage. | + | To still save the ship while charging the central component minimally, one simply needs to solve the quadratic equation based on the inequality above: $50 \text{ C} \cdot q_{comp} - q_{comp}^2 = 530 \text{ C}^2$. An application of the quadratic |
- | \\ $*$Note about $\Delta U$: We include a $1/x_i$ term, which we know is very small, and will not contribute to the change in electric potential energy. Technically, | + | <WRAP TIP> |
+ | ===$*$Note about $\Delta U$=== | ||
+ | We include a $1/x_i$ term, which we know is very small, and will not contribute to the change in electric potential energy. Technically, | ||
+ | </ |