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184_notes:examples:week3_spaceship_asteroid [2018/02/03 20:06] – [Solution] 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=== | ||
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The asteroid' | The asteroid' | ||
- | + | <WRAP TIP> | |
- | ===Approximations & Assumptions=== | + | === Assumptions === |
- | | + | We did not include |
* 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, | ||
+ | </ | ||
- | 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} \\ | ||
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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 |
<WRAP TIP> | <WRAP TIP> |