A spacecraft is on a mission to reach Mars, which is approximately $78$ million kilometers away from Earth. If the spacecraft travels at a constant speed of $25,000$ km$/$h$,$ how long will it take to reach Mars? Additionally, considering the gravitational influence of both Earth and Mars, determine the changes in velocity required to enter Mars' orbit from the spacecraft's trajectory.

Calculate the total energy required to melt $1000$ kg of ice at $-10$C$,$ heat the water to $100$C$,$ and then convert it to steam. Assume specific heat capacities of ice, water, and steam as $2\mathrm{.}1$ kJ$/$kg$$C$,4\mathrm{.}18$ kJ$/$kg$$C$,$ and $2\mathrm{.}0$ kJ$/$kg$$C$,$ respectively, and the latent heats of fusion and vaporization of water are $334$ kJ$/$kg and $2260$ kJ$/$kg$,$ respectively.

A cylindrical tank with a height of $10$ meters and a diameter of $5$ meters is being filled with water at a rate of $50$ liters per second. Determine the time required to fill the tank. Also, calculate the pressure exerted at the bottom of the tank when it is full, considering the density of water as $1000$ kg$/$m$^3$ and the acceleration due to gravity as $9\mathrm{.}8$ m$/$s$^2.$

In a double$-$slit experiment, monochromatic light of wavelength $600$ nm is used. The slits are separated by a distance of $0\mathrm{.}1$ mm$,$ and the screen is placed $1$ meter away from the slits. Calculate the distance between the central maximum and the first$-$order maximum on the screen. Additionally, derive the expression for the intensity distribution on the screen.

A $10$ kg block slides down a frictionless incline that makes an angle of $30$ degrees with the horizontal. Determine the acceleration of the block and the time it takes to slide down a distance of $5$ meters along the incline. If a frictional force of $20$ N is introduced, calculate the new acceleration and time.

A solenoid with $500$ turns and a length of $0\mathrm{.}5$ meters carries a current of $2$ A$.$ Calculate the magnetic field inside the solenoid. If an iron core with a relative permeability of $1000$ is inserted into the solenoid, determine the new magnetic field inside the solenoid.

A planet orbits a star in an elliptical orbit with a semi$-$major axis of $1$ AU and an eccentricity of $0\mathrm{.}1.$ Using Kepler$$s laws, calculate the period of the planet$$s orbit. Also, determine the closest and farthest distances of the planet from the star in its orbit.

In an RLC circuit with a resistance of $10,$ an inductance of $0\mathrm{.}05$ H$,$ and a capacitance of $100$ F$,$ determine the resonant frequency. Calculate the impedance of the circuit at resonance and the quality factor of the circuit. Additionally, analyze the transient response of the circuit when a step voltage is applied.

A satellite in a geostationary orbit around Earth has an orbital radius of approximately $42,164$ km$.$ Calculate the orbital speed of the satellite. Determine the gravitational force acting on the satellite, given its mass is $500$ kg$.$ Compare the required centripetal force to the gravitational force to confirm the satellite's stable orbit.

A chemical reaction has a rate constant that follows the Arrhenius equation:

$$

$=$

$$

exp

$($

$$

$$

$$

$/$

$$

$$

$)$

k$=$Aexp$($E

a

$/$RT$).$ Given that the rate constant doubles when the temperature increases from $300$ K to $310$ K$,$ determine the activation energy

$$

$$

E

a

of the reaction. If the pre$-$exponential factor

$$

A is known, calculate the rate constant at $350$ K$.$

A spherical capacitor consists of two concentric spherical conductors with radii

$$

$1$

R

$1$

and

$$

$2$

R

$2$

$.$ Derive the expression for the capacitance of the spherical capacitor. If

$$

$1$

$=$

$1$

R

$1$

$=1$ cm and

$$

$2$

$=$

$3$

R

$2$

$=3$ cm$,$ calculate the capacitance. Determine the electric field at a distance

$$

r from the center, where

$$

$1$

$<$

$$

$<$

$$

$2$

R

$1$

What is thread in computer science?