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184_notes:examples:week3_spaceship_asteroid [2017/09/01 18:18] – [Solution] tallpaul | 184_notes:examples:week3_spaceship_asteroid [2021/05/19 14:37] – schram45 | ||
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+ | [[184_notes: | ||
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=====Example: | =====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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"The asteroid is coming from the due starboard direction with respect to the ship. The asteroid is approaching with a speed of 4,000 meters/ | "The asteroid is coming from the due starboard direction with respect to the ship. The asteroid is approaching with a speed of 4,000 meters/ | ||
- | You get to thinking, and you remember there is another set of controls in an unlocked room. This set of controls is designed specifically to prevent asteroid collisions. The ship uses an advanced long-distance wiring setup to extract charge from the asteroid and bring it to a component located at the center of the ship. By charging the component, a repulsive electric force is generated between the asteroid and the component. Is it possible to charge the component in this way to prevent a collision? If so, what is the minimum amount of charge | + | You get to thinking, and you remember there is another set of controls in an unlocked room. This set of controls is designed specifically to prevent asteroid collisions. The ship uses an advanced long-distance wiring setup to extract charge from the asteroid and bring it to a component located at the center of the ship. By charging the component, a repulsive electric force is generated between the asteroid and the component. Is it possible to charge the component in this way to prevent a collision? If so, what is the minimum amount of charge |
- | + | ||
- | {{ 184_notes: | + | |
===Facts=== | ===Facts=== | ||
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* The asteroid is approaching from the starboard (right) direction. | * The asteroid is approaching from the starboard (right) direction. | ||
* The central component can be charged using charge from the asteroid. | * The central component can be charged using charge from the asteroid. | ||
+ | * The electric potential energy of a point charge in the electric field of another point charge is $$U_r=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}$$ This was derived in the notes [[184_notes: | ||
- | ===Lacking=== | + | ===Assumptions=== |
- | * The current distance between | + | * Asteroid and Central Component are point charges: |
- | * The distribution of charge on the asteroid. | + | * Asteroid is going straight towards |
- | * The distribution of charge on the central component | + | * Ship is a rectangle: Creates a simplified model for the ship, and is a worse case scenario for the ship shape. |
+ | |||
+ | ===Goal=== | ||
+ | * Prevent the asteroid collision using the long-distance wiring setup. | ||
- | ===Approximations | + | ===Representations=== |
+ | {{ 184_notes: | ||
+ | <WRAP TIP> | ||
+ | === Approximations === | ||
+ | We want to make a useful representation, | ||
* 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 the system of the ship and the asteroid is closed, i.e., the energy | + | These are all reasonable approximations, |
- | * The ship is currently floating through space, | + | </ |
- | * The ship is far more massive than the asteroid | + | |
- | + | ||
- | ===Representations=== | + | |
- | * We represent | + | |
- | * We represent the example and its solution | + | |
- | * It will help to draw a representation of what we have decided about the problem above. We show our representation below. | + | |
{{ 184_notes: | {{ 184_notes: | ||
====Solution==== | ====Solution==== | ||
- | We choose to solve this example using energy. The system is the asteroid and ship with nothing in the surroundings, so energy | + | This is a complicated problem. |
- | 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{ | + | <WRAP TIP> |
+ | === Plan === | ||
+ | We will use conservation of energy to find the distance the asteroid can reach. We'll go through the following steps. | ||
+ | * 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, and an equivalent decrease in kinetic 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 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{ | ||
\begin{align*} | \begin{align*} | ||
- | \Delta U & | + | \Delta U &= \frac{1}{4\pi\epsilon_0}\frac{q_{ast}q_{comp}}{x_f} |
- | & | + | & |
- | & | + | |
- | & | + | |
- | & | + | |
- | & | + | |
\end{align*} | \end{align*} | ||
- | FIXME It's probably ok to just use the initial and final energy for point charge here rather than doing the integral again | + | Ultimately, we want to figure out what $q_{comp}$ needs to be. $q_{ast}$ also depends on this, so let's solve for $q_{comp}q_{ast}$, |
- | + | ||
- | Ultimately, we want to figure out what $q_{comp}$ needs to be. $q_{ast}$ also depends on this, so let's solve for $q_{comp}q_{ast}$, | + | |
\begin{align*} | \begin{align*} | ||
0 &= \Delta U + \Delta K \\ | 0 &= \Delta U + \Delta K \\ | ||
- | &= \frac{q_{comp}q_{ast}}{4\pi\epsilon_0}\left(\frac{1}{x_i}-\frac{1}{x_f}\right) -\frac{1}{2}mv^2 \\ | + | &= \frac{q_{comp}q_{ast}}{4\pi\epsilon_0}\left(\frac{1}{x_f}-\frac{1}{x_i}\right) -\frac{1}{2}mv^2 \\ |
&< \frac{q_{comp}q_{ast}}{4\pi\cdot 8.85\cdot 10^{-12}\frac{\text{C}^2}{\text{Jm}}}\left(\frac{1}{30 \text{ m}}-\frac{1}{2.4\cdot 10^6 \text{ m}}\right) -\frac{1}{2}\left(20000 \text{ kg}\cdot\left(4000 \text{ m/ | &< \frac{q_{comp}q_{ast}}{4\pi\cdot 8.85\cdot 10^{-12}\frac{\text{C}^2}{\text{Jm}}}\left(\frac{1}{30 \text{ m}}-\frac{1}{2.4\cdot 10^6 \text{ m}}\right) -\frac{1}{2}\left(20000 \text{ kg}\cdot\left(4000 \text{ m/ | ||
&= \frac{q_{comp}q_{ast}}{\text{C}^2}\cdot 3.00\cdot 10^8 \text{ J} - 1.6\cdot 10^{11} \text{ J} | &= \frac{q_{comp}q_{ast}}{\text{C}^2}\cdot 3.00\cdot 10^8 \text{ J} - 1.6\cdot 10^{11} \text{ J} | ||
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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 $1/x_f$ 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, | ||
+ | </ |