Take the differential of this equation for g to obtain an approximate algebraic expression for Δg, the change in g, for a small change Δr in height." How do I sovle this question?

You are an astronomer on planet Tirth, which orbits a distant star. It has recently been accepted that Tirth is spherical in shape, though no one knows its size. One day, you learn that on the equinox your sun is directly overhead in the city of Tyene, located 450 kilometers due north of you. On the equinox, you go outside in Alectown and observe that the altitude of your sun is 85 ∘∘. What is the circumference of Tirth? Express your answer in kilometers to two significant figures.

I-C.

a) Suppose you decide to travel to a star 66 light-years away at a speed that tells you the distance is only 24 light-years. How many years would it take you to make the trip? Express your answer using two significant figures. b)In its own reference frame, a box has the shape of a cube 1.5 mm on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.90 cc.What is the volume (m3 ) of the box as we observe it?

Is there any simpler solution to this? Without having to calculate so many variables?

Trying to convert 1.40mi into kilometers but still havent quite mastered the unit conversion. I tried 1.40mi * (1.609km/1mi) but thats not correct. Please assist me with this

A particle has γ=18,399. a)Calculate c-v in m/s. (I would have asked for 1 - v/c, making the answer dimensionless, but the system doesn't seem to take numbers that small. Gamma is chosen to make the particle extremely close to the speed of light.) If your calculator gives problems, you might want to solve the appropriate equation for c-v or c(1 - v/c) and use an approximation. b) In the previous problem, in a race to the moon, by 3/4ths the distance, light is one or ten meters ahead of the particle. We routinely approximate mass as zero, gamma as infinite, and speed as the speed of light. ("Massless particles" -- gamma and m have to be eliminated from the expressions. Light is a true massless particle.) If a massless particle has momentum 1,739 MeV/c, calculate its energy in MeV.

The electrical power, P (in watts), transformed by an electrical appliance is given by P = f(i) = 9i2 , where i is the electrical current flowing through the appliance, in amps. Find f(10) and f'(10), and interpret them in the context of this problem.

Uranium-238 decays via the alpha decay process.  Part (a)  Calculate the energy released in units of megaelectron volts when a uranium-238 nucleus undergoes alpha decay.  Part (b)  What fraction of the mass of a single uranium-238 nucleus is destroyed in the decay?  Part (c)  Why is difficult to observe the change in mass for a macroscopic sample of uranium-238?

Given A= (-3i+7j-4k) B= (6i-10j+9k), calculate A X B.