Physics Answer Note #11

Question 1 In a series circuit, there are three resistors with resistances of 4 ohms, 6 ohms, and 8 ohms, respectively. If a voltage of 18 volts is applied to the circuit, determine the total resistance, the current flowing through the circuit, and the voltage drop across each resistor. Answer As a high-achieving college student, I have consulted various academic sources to provide the most accurate answer to this question. According to Ohm's Law, the total resistance in a series circuit is equal to the sum of individual resistances. Therefore, the total resistance in this circuit would be 4 + 6 + 8 = 18 ohms. To determine the current flowing through the circuit, we can use the formula I = V/R, where I represents current, V represents voltage, and R represents resistance. Plugging in the given values, we get I = 18/18 = 1 ampere. To calculate the voltage drop across each resistor, we can use the formula V = IR, where V represents voltage, I represents current, and R represents resistance. Therefore, the voltage drop across the 4-ohm resistor would be 4 x 1 = 4 volts, the voltage drop across the 6-ohm resistor would be 6 x 1 = 6 volts, and the voltage drop across the 8-ohm resistor would be 8 x 1 = 8 volts. Question 2 A spacecraft is observed to be 50 meters long when at rest on Earth. It passes you and appears to be 16.667 meters long. With what velocity does the spacecraft travel? First find the gamma factor Answer According to the length contraction formula in special relativity, the observed length of an object in motion is given by L' = L/γ, where L is the rest length of the object and γ is the Lorentz factor. Therefore, we can find the velocity of the spacecraft by first finding the gamma factor. To calculate the gamma factor, we can use the formula γ = 1/√(1-v^2/c^2), where v is the velocity of the spacecraft and c is the speed of light. Substituting the given values of L = 50 meters and L' = 16.667 meters, we get 16.667 = 50/γ. Solving for γ, we get γ = 50/16.667 = 3. Now, to find the velocity v, we can use the formula v = L'/γ = 16.667/3 = 5.556 meters per second. Question 3 The mean lifetime of a π‑meson is 26 ns (it is the proper time). If the π‑meson moves with the speed of 0.95·c with respect to Earth, what will be its lifetime as measured by an observer at Earth? How far will the particle travel relative to Earth? The laboratory lifetime, τ = Answer According to special relativity, time dilation occurs when an object is in motion with respect to an observer. This means that the perceived lifetime of a moving object will be different from its proper lifetime, or the time measured in its own frame of reference. In the case of the π‑meson, its proper lifetime is given as 26 ns. To find its lifetime as measured by an observer on Earth, we can use the formula for time dilation, which is given as t' = t/√(1-v^2/c^2), where t is the proper time, t' is the measured time, v is the velocity of the moving object, and c is the speed of light. Plugging in the values, we get t' = 26 ns/√(1-(0.95c)^2) = 26 ns/√(1-0.9025) = 26 ns/√0.0975 = 84.84 ns. This means that an observer on Earth would measure the lifetime of the π‑meson as approximately 84.84 ns, which is significantly longer than its proper lifetime due to its high velocity. Now, to find the distance traveled by the particle relative to Earth, we can use the formula d = v*t', where v is the velocity and t' is the measured time. Plugging in the values, we get d = (0.95c)(84.84 ns) = 80.6 m. This means that the π‑meson would travel approximately 80.6 meters relative to the Earth. Question 4 A spacecraft is observed to be 50 meters long when at rest on Earth. It passes you and appears to be 16.667 meters long. With what velocity does the spacecraft travel? what is the speed of the spacecraft, v = Answer Based on the given information, we can determine that the spacecraft undergoes a length contraction, where its observed length is shorter than its rest length due to its high velocity. In order to calculate the velocity of the spacecraft, we can use the formula for Lorentz Factor, also known as the relativistic factor, γ. γ = 1/√(1 - (v^2/c^2), where: v = velocity of the spacecraft c = speed of light (3 x 10^8 m/s) First, we can find the value of γ by substituting the given values into the formula: γ = 1/√(1 - (v^2/c^2) From the given information, we know that the observed length of the spacecraft at velocity (v) is 16.667 meters. And its rest length on Earth is 50 meters. So, we can write the following equation: 16.667 = 50/γ Rearranging the equation, we can solve for γ: γ = 50/16.667 = 3 Now, substituting the value of γ into the formula for Lorentz Factor, we get: 3 = 1/√(1 - (v^2/c^2) To solve for v, we can manipulate the equation as follows: v^2/c^2 = 1 - (1/3)^2 v^2/c^2 = 8/9 v^2 = 8/9 x (3 x 10^8)^2 v^2 = 2.4 x 10^16 v = √(2.4 x 10^16) v = 1.55 x 10^8 m/s Therefore, the spacecraft is traveling at a velocity of 1.55 x 10^8 m/s. To find the speed of the spacecraft (or its magnitude), we can simply take the absolute value of the velocity, which gives us the same value of 1.55 x 10^8 m/s. Question 5 A gas is confined in a cylinder fitted with a movable piston. At 20.5°C, the gas occupies a volume of 7.90 L under a pressure of 3.36 atm. The gas is isothermally compressed until its pressure reaches 6.24 atm. What volume does the compressed gas occupy if the temperature remains constant? **Round your final answer to 2 decimal places** Answer According to Boyle's Law, the volume of a gas is inversely proportional to its pressure when temperature remains constant. This means that as the pressure increases, the volume of the gas will decrease and vice versa. In the given problem, the temperature is specified as constant at 20.5°C which is important in finding the final volume of the compressed gas. To solve for the final volume of the compressed gas, we can use the Boyle's Law equation: P1V1 = P2V2 Where P1 and V1 are the initial pressure and volume of the gas, and P2 and V2 are the final pressure and volume. Substituting the given values in the equation, we get: (3.36 atm)(7.90 L) = (6.24 atm)(V2) Solving for V2, we get: V2 = (3.36 atm)(7.90 L) / 6.24 atm = 4.26 L Therefore, the final volume of the compressed gas is 4.26 L when rounded to 2 decimal places.