Q7

Charging the Capacitor Having stabilized the fuel, you look to your computer for next steps. You are told that the spacecraft uses a "start capacitor" that must reach a required voltage in order to start its engine. This voltage is determined by the fixed capacitance and charge as a function of time, q(t). These capacitors are charged and stored in your current location and must be brought to the spacecraft. Your computer provides you with the following definitions: Charge: The charge as a function of time is g(t) = [i(t) dt, where 7 (t) = 48t7 + 12t3 8 is the current as a function of time. You need to provide the computer with the function g () in order for it to appropriately charge the start capacitor. If an unknown constant is needed, it can be included using "+ C", being sure to use a capital letter as the lowercase is used for capacitance. q(t) = BEM The Spacecraft Password Entering the room, you see the spacecraft. Finally, you will be able to leave this planet! Upon closer inspection, you realize that there is a password to enter the ship. Your computer translates the password as follows: Find all functions whose derivative is 3cos (). Your computer is able to provide a hint that these functions only differ by a constant and that an arbitrary constant can be included using "+ C". In order to appropriately enter the password, you will need to enclose arguments of functions in parentheses. For example, sin (233) Spacecraft Password: B The Computer Password As you enter the spacecraft, you look to connect your computer To your dismay, the controls are also password protected. Your computer helpfully translates the prompt and notifies you that a three-digit number is needed. Your computer is able to determine the algorithm for determining each number, along with some password hints. The computer displays the below information- Password Hints: 6 [ fla)dz1 4 6 [ g(z)dz =18 4 [a(z)dz =9 6 First Number: f4(3 f(z) 3g(x))dz 4 6 second Number: [ (3 f(z) 3g(z))dz + [ f(z) dz 6 6 6 Third Number: [ 3g(z)dz 4 First Number Second Number Third Number Number Number Number Plotting the Course With access to the spacecraft's controls, you connect your computer and begin installing the fuel tank and start capacitor. Your computer informs you that it can plot most of the course to your home planet but needs assistance with determining your velocity and vertical position at a critical moment in time. Your computer provides you with a function that models your acceleration during the first few seconds of flight. You will need to take this information and calculate your velocity and vertical position at time = 2 s. Your computer notes that you can use the below initial conditions in supporting your calculations: Initial time: 0 s Initial vertical position: 0 m Initial velocity: 0 m/s Acceleration during this time: a (t) = 443 + 47t mis? What values (to 2 decimal places) do you provide your computer? v(2) = | Number mis 8 (2) = | number m Connecting the Start Capacitor The spacecratit is nearly ready for takeoffl The last task is {o attach the start capacitor to the ship in a simple RC-circuit. Your computer warns you that some adjustments may need to be made if the heat generated in the resistor over the initial time interval is higher than a given threshold. Your computer is able to provide the following information, along with a note that the units are already accounted for in the equations: V.2 g Heat Generated in Resistor over the first 3 seconds: @ = %fD e_2t/(RC) dt joules Voltage of the Battery, Vb = 3valts Resistance: R = 3 ohms Capacitance: C' = % farads Maximum Heat Allowed in Resistor: Q) ax 1.5 joules Is the heat generated in the resistor over the first 3 seconds below the maximum allowable threshold? 0 Yes O No What is the difference between the maximum threshold (provided above by your computer) and the amount of heat generated in the resistor (that you calculated)? Provide a positive value up to three decimal places. Number joules