HSC Physics Does ∆U always equal mg∆h?

A tutorial sheet of gravitational potential energy questions is below.

1. What do the symbols in the equation ∆U=mg∆h mean?
2. When is it correct to use the equation ∆U=mg∆h?
3. What is the meaning of the equation U=-GMm/r?
4. When should we use the equation U=-GMm/r?
5. An object of mass m is lifted through a vertical height ∆h from the surface of the Earth. If the radius of the Earth is R shew that ∆U=mg∆hR/(R+h), where g is the acceleration due to gravity at the surface of the Earth.
6. An apple of mass 85 g falls from a height of 2.5 m to the ground. What is the change in gravitational potential energy of the apple? [0.99999961mg∆h ≅ mg∆h, -2.08 J]
7. Mt Everest is 8848 m sbove sea level. If the radius of the Earth is 6400 km show that the increase in gravitational potential energy of a climber of mass m in going from sea level to the top of Mt Everest is 0.9986mg∆h.
8. The International Space Station (ISS) orbits the Earth at an average altitude of 408 km. If its mass is 419,700 kg determine the gain in gravitational potential energy in lifting this mass from the surface of the Earth to its final height.[0.9401mg∆h, 1.58x10¹² J]

IB HL Physics Resolution

Image resolution is a poorly understood topic in IB HL Physics. A tutorial sheet is given below.

1. A double star is examined through a telescope. The stars appear too close together to be resolved. To resolve the image of the stars a coloured filter is placed in front of the telescope. Which coloured filter could allow the images to be resolved? Red or violet?
2. Two stars are observed through a telescope and appear too close together to be resolved. Does increasing the magnification of the telescope resolve the images?
3. The headlights of a stationary car subtend an angle of 1" at a distant point. The human eye can distinguish between two images that are 1' apart. If the wavelength of the light is 550 nm, determine the minimum diameter of a telescope that can allow the headlights to be resolved as separate images.
4. A satellite is in orbit 350 km above the surface of the Earth. Newspint has a size of 3.0 mm. Determine the diameter of the aperture of a camera in the satellite that will allow a newspaper on the Earth to be read from orbit in light of wavelength (a) 600 nm, (b) 400 nm. The resolution of the human eye is 1'.
5. If we increase the amount of light entering a telescope, keeping the diameter and the wavelength constant, does this improve the image resolution? Why?