🔑:## Step 1: Calculate the initial potential energy of the asteroidThe initial potential energy (U_i) of the asteroid can be calculated using the formula U = -G * (M_E * m) / r, where G is the gravitational constant, M_E is the mass of the Earth, m is the mass of the asteroid, and r is the distance from the center of the Earth to the asteroid. Given that the asteroid is 3 Earth radii away, the initial distance r_i = 3 * R_E, where R_E is the radius of the Earth.## Step 2: Plug in the values to calculate the initial potential energyU_i = -G * (M_E * m) / (3 * R_E) = -6.67 × 10^–11 N m^2 kg^–2 * (5.98 × 10^24 kg * 2.6 × 10^6 kg) / (3 * 6.38 × 10^6 m).## Step 3: Calculate the initial potential energyU_i = -6.67 × 10^–11 * 5.98 × 10^24 * 2.6 × 10^6 / (3 * 6.38 × 10^6) = -6.67 × 10^–11 * 15.5088 × 10^30 / (19.14 × 10^6) = -6.67 × 10^–11 * 15.5088 × 10^30 / 19.14 × 10^6 = -5.313 × 10^13 J.## Step 4: Calculate the initial kinetic energy of the asteroidThe initial kinetic energy (K_i) of the asteroid can be calculated using the formula K = 0.5 * m * v^2, where m is the mass of the asteroid and v is its initial speed.## Step 5: Plug in the values to calculate the initial kinetic energyK_i = 0.5 * 2.6 × 10^6 kg * (8.7 × 10^3 m/s)^2 = 0.5 * 2.6 × 10^6 * 75.69 × 10^6 = 0.5 * 197.394 × 10^12 = 98.697 × 10^12 J.## Step 6: Calculate the total initial energy of the asteroidThe total initial energy (E_i) is the sum of the initial potential energy and the initial kinetic energy, E_i = U_i + K_i.## Step 7: Plug in the values to calculate the total initial energyE_i = -5.313 × 10^13 J + 98.697 × 10^12 J = -5.313 × 10^13 J + 0.98697 × 10^13 J = -4.32603 × 10^13 J.## Step 8: Calculate the final potential energy of the asteroid when it hits the groundThe final potential energy (U_f) of the asteroid when it hits the ground can be calculated using the formula U = -G * (M_E * m) / r, where r is now the radius of the Earth (R_E) since the asteroid is at the surface.## Step 9: Plug in the values to calculate the final potential energyU_f = -G * (M_E * m) / R_E = -6.67 × 10^–11 N m^2 kg^–2 * (5.98 × 10^24 kg * 2.6 × 10^6 kg) / (6.38 × 10^6 m).## Step 10: Calculate the final potential energyU_f = -6.67 × 10^–11 * 5.98 × 10^24 * 2.6 × 10^6 / (6.38 × 10^6) = -6.67 × 10^–11 * 15.5088 × 10^30 / 6.38 × 10^6 = -15.939 × 10^13 J.## Step 11: Calculate the final kinetic energy of the asteroid when it hits the groundSince energy is conserved, the total final energy (E_f) equals the total initial energy (E_i), and E_f = U_f + K_f, where K_f is the final kinetic energy. Rearranging for K_f gives K_f = E_i - U_f.## Step 12: Plug in the values to calculate the final kinetic energyK_f = E_i - U_f = -4.32603 × 10^13 J - (-15.939 × 10^13 J) = -4.32603 × 10^13 J + 15.939 × 10^13 J = 11.613 × 10^13 J.The final answer is: boxed{1.1613times10^{14}}

🔑:Replacing an O2 sensor on a 1999 Ford Contour is a relatively straightforward process that can be done with basic tools. Here's a step-by-step guide to help you replace the O2 sensor:Location of the O2 Sensor:The O2 sensor is located in the exhaust system, near the catalytic converter. On a 1999 Ford Contour, there are two O2 sensors: one upstream (before the catalytic converter) and one downstream (after the catalytic converter). The upstream O2 sensor is located on the driver's side of the exhaust system, about 6-8 inches from the exhaust manifold. The downstream O2 sensor is located on the passenger's side of the exhaust system, about 6-8 inches from the catalytic converter.Tools Required:* O2 sensor socket (32 mm or 1 1/16 inch)* Ratchet and extension* Torx screwdriver (for removing heat shield)* Pliers or wrench (for removing O2 sensor wiring connector)* New O2 sensor (make sure it's compatible with your vehicle)Steps Involved in the Replacement Process:1. Warm up the engine: Drive the vehicle for a few minutes to warm up the engine. This will help loosen the O2 sensor and make it easier to remove.2. Locate the O2 sensor: Identify the O2 sensor you want to replace (upstream or downstream). You'll see a wire connected to the sensor, and a heat shield covering the sensor.3. Remove the heat shield: Use a Torx screwdriver to remove the screws holding the heat shield in place. Gently pull the heat shield away from the O2 sensor.4. Disconnect the O2 sensor wiring connector: Use pliers or a wrench to loosen the nut holding the wiring connector to the O2 sensor. Pull the connector off the sensor.5. Remove the O2 sensor: Use an O2 sensor socket to loosen the sensor. Turn the sensor counterclockwise until it's loose, then pull it out of the exhaust system.6. Install the new O2 sensor: Insert the new O2 sensor into the exhaust system, making sure it's properly seated. Turn the sensor clockwise until it's snug.7. Tighten the O2 sensor: Use the O2 sensor socket to tighten the sensor in a clockwise direction. Make sure it's snug, but don't overtighten.8. Reconnect the O2 sensor wiring connector: Connect the wiring connector to the new O2 sensor. Tighten the nut holding the connector in place.9. Replace the heat shield: Put the heat shield back in place, and secure it with the Torx screws.10. Clear any trouble codes: Use a code scanner to clear any trouble codes related to the O2 sensor.11. Test the vehicle: Start the engine and check for any trouble codes or symptoms related to the O2 sensor.Tips and Precautions:* Make sure to purchase a new O2 sensor that's compatible with your vehicle.* Use a torque wrench to tighten the O2 sensor to the specified torque (usually around 30-40 ft-lbs).* Be careful not to touch the O2 sensor's tip, as the oils from your skin can contaminate the sensor and affect its performance.* If you're not comfortable with this replacement, consider consulting a professional mechanic.I hope this helps! If you have any further questions or concerns, feel free to ask.

🔑:## Step 1: Determine the total resistance of the parallel resistors (25, 23, 34, and 45 ohms)To find the total resistance of the resistors in parallel, we use the formula for parallel resistances: ( frac{1}{R_{total}} = frac{1}{R_1} + frac{1}{R_2} + frac{1}{R_3} + frac{1}{R_4} ). Substituting the given values, we get ( frac{1}{R_{total}} = frac{1}{25} + frac{1}{23} + frac{1}{34} + frac{1}{45} ).## Step 2: Calculate the total resistance of the parallel resistorsLet's calculate ( frac{1}{R_{total}} ) for the parallel part: ( frac{1}{R_{total}} = frac{1}{25} + frac{1}{23} + frac{1}{34} + frac{1}{45} ). This equals ( frac{1}{R_{total}} = 0.04 + 0.04348 + 0.02941 + 0.02222 ), which simplifies to ( frac{1}{R_{total}} = 0.13511 ). Therefore, ( R_{total} = frac{1}{0.13511} approx 7.4 ) ohms.## Step 3: Calculate the total resistance of the entire circuitThe total resistance (( R_{circuit} )) of the circuit is the sum of the resistances of the resistors in series plus the total resistance of the parallel part. So, ( R_{circuit} = 12 + 7.4 + 56 ).## Step 4: Calculate the total resistance valueSubstituting the values, we get ( R_{circuit} = 12 + 7.4 + 56 = 75.4 ) ohms.## Step 5: Apply Ohm's Law to find the total current in the circuitUsing Ohm's Law, ( I = frac{V}{R} ), where ( V ) is the total voltage (200 V) and ( R ) is the total resistance of the circuit (75.4 ohms), we find ( I = frac{200}{75.4} ).## Step 6: Calculate the total currentCalculating the total current gives ( I approx 2.65 ) amps.## Step 7: Determine the voltage across the parallel resistorsSince the parallel resistors are in series with the 12 and 56 ohm resistors, we can use the total current to find the voltage drop across the 12 ohm resistor and then across the parallel part. The voltage drop across the 12 ohm resistor is ( V_{12} = I times R = 2.65 times 12 ).## Step 8: Calculate the voltage drop across the 12 ohm resistor( V_{12} = 2.65 times 12 approx 31.8 ) volts.## Step 9: Calculate the voltage drop across the parallel resistors and the 56 ohm resistorThe total voltage drop across the parallel part and the 56 ohm resistor is ( 200 - 31.8 = 168.2 ) volts. This voltage is divided between the parallel resistors and the 56 ohm resistor.## Step 10: Determine the voltage drop across the parallel resistorsSince the parallel resistors have a total resistance of 7.4 ohms, and they are in series with the 56 ohm resistor, we need to find the voltage drop across the parallel resistors using the ratio of resistances or by applying the current through the parallel part.## Step 11: Calculate the voltage drop across the parallel resistorsGiven that the current through the parallel part is the same as the total current (since it's in series with the other resistors), we can calculate the voltage drop across the parallel resistors using ( V_{parallel} = I times R_{parallel} = 2.65 times 7.4 ).## Step 12: Perform the calculation for the voltage drop across the parallel resistors( V_{parallel} = 2.65 times 7.4 approx 19.61 ) volts.## Step 13: Determine the voltage drop across the 56 ohm resistorThe voltage drop across the 56 ohm resistor can be found by subtracting the voltage drop across the parallel resistors from the total voltage drop across the parallel part and the 56 ohm resistor: ( V_{56} = 168.2 - 19.61 ).## Step 14: Calculate the voltage drop across the 56 ohm resistor( V_{56} = 168.2 - 19.61 approx 148.59 ) volts.## Step 15: Verify the calculations for the voltage dropsTo ensure accuracy, we verify that the sum of the voltage drops equals the total voltage: ( V_{12} + V_{parallel} + V_{56} = 31.8 + 19.61 + 148.59 ).## Step 16: Perform the verification calculation( 31.8 + 19.61 + 148.59 = 200 ) volts, which matches the given total voltage.The final answer is: boxed{2.65}

🔑:Population inversion is a crucial condition for amplification in laser operation, where the number of atoms or molecules in an excited state is greater than the number in a lower energy state. This is necessary for stimulated emission, which is the process by which an excited atom or molecule releases a photon, inducing other nearby atoms or molecules to release photons of the same energy and phase.Why population inversion is required:In a normal population, the number of atoms or molecules in a lower energy state is greater than the number in an excited state. When a photon interacts with an atom or molecule in the lower energy state, it is more likely to be absorbed, leading to an increase in the population of the excited state. However, this process does not lead to amplification, as the absorbed photon is not released again.In contrast, when a photon interacts with an atom or molecule in an excited state, it can stimulate the release of another photon of the same energy and phase, a process known as stimulated emission. This process leads to amplification, as the released photon can stimulate other nearby atoms or molecules to release photons, creating a cascade of photons.However, for stimulated emission to dominate over absorption, the population of the excited state must be greater than the population of the lower energy state, i.e., a population inversion must exist.Mechanisms for achieving population inversion:Different types of lasers use various mechanisms to achieve population inversion:1. Ruby lasers: Ruby lasers use a three-level system, where a pump source excites the chromium ions in the ruby crystal to a higher energy state. The chromium ions then decay to a metastable state, which has a longer lifetime than the higher energy state. The population inversion is achieved by pumping the chromium ions to the higher energy state faster than they can decay to the metastable state.2. He-Ne lasers: He-Ne lasers use a four-level system, where a discharge excites the helium atoms to a higher energy state. The excited helium atoms then collide with neon atoms, transferring their energy and exciting the neon atoms to a higher energy state. The population inversion is achieved by the selective excitation of the neon atoms, which have a longer lifetime in the excited state than the helium atoms.3. Diode lasers: Diode lasers use a semiconductor material with a p-n junction. When a forward bias is applied to the p-n junction, electrons from the n-side and holes from the p-side recombine, releasing energy in the form of photons. The population inversion is achieved by the injection of electrons and holes into the active region, where they recombine to form a population of excited states.Role of stimulated emission:Stimulated emission is the key process that enables amplification in laser operation. When a photon interacts with an atom or molecule in an excited state, it can stimulate the release of another photon of the same energy and phase. This process leads to amplification, as the released photon can stimulate other nearby atoms or molecules to release photons, creating a cascade of photons.In summary, population inversion is required for amplification in laser operation because it allows stimulated emission to dominate over absorption. Different types of lasers use various mechanisms to achieve population inversion, including three-level and four-level systems, and semiconductor materials with p-n junctions. Stimulated emission is the key process that enables amplification, and it is only possible when a population inversion exists.