In the action adventure film indiana jones and the raiders of the lost ark, indiana jones searches for the ark of the covenant. in one well-known nail biting scene indiana tries to escape being crushed by a huge boulder rolling down a sloping terrain inside a tunnel. the rolling boulder of mass m rolls down the only path that indiana jones can take. assume that the friction force between the sphere and the surface of the hill is negligible. at a particular time t1 the spherical boulder is at location 1 and moving with velocity v1. at a later time t2 the sphere is at location 2 and moving with velocity v2. locations 1 and 2 are at vertical heights h1 and h2 respectively, from the bottom of the tunnel. when answering the following questions use 1 and 2 for subscripts to identify the different variables at the two different locations. part (a): u1 = mgh1part (b): what is the equation for translational kenetic energy, kt,1 of the spherical boulder? kt,1 = .5mv12part (c): what is the equation for the rotational kinetic energy kr,1of the spherical boulder? kr,1 = .5w12 ; kr,1 = (1/5)mv12part (d): what is the equation for the total kinetic energy ktotal,1of the spherical boulder in terms of its linear speed? ktotal,1 = (7/10)mv12part (e): you can write a similar expression for the potential energy u2 and the total kinetic energy ktotal,2 of the spherical boulder when it is at location 2.a reasonable choice for the initial location 1 would be the point where the spherical boulder starts from rest; i. e. where the sphere has zero initial velocity. use this fact and your answers to parts (a), (b), (c), and (d) to write the mathematical equation for the conservation of mechanical energy in terms of the mass, velocity, height and g. keep the kinetic energy terms on the left and the potential energy terms on the right. include only non-zero terms. (use any variable or symbol stated above along with the following as necessary: g.)

  sequence  is a continuous and distinctive band of  stars  that appears on plots of stellar  color  versus  brightness. these color-magnitude plots are known as  hertzsprung–russell diagramsafter their co-developers,  ejnar hertzsprungand  henry norris russell. stars on this band are known as  main-sequence stars  or  dwarf stars. these are the most numerous true stars in the universe, and include the earth's  sun.after condensation and ignition of a star, it generates  thermal energy  in its dense core region through  nuclear fusion  of  hydrogen  into  helium. during this stage of the star's lifetime, it is located on the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and age. the cores of main-sequence stars are in  hydrostatic equilibrium, where outward thermal pressure from the hot core is balanced by the inward pressure of  gravitational collapse  from the overlying layers. the strong dependence of the rate of energy generation on temperature and pressure to sustain this balance. energy generated at the core makes its way to the surface and is radiated away at the  photosphere. the energy is carried by either  radiation  or  convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both.the main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. stars below about 1.5 times the  mass of the sun  (1.5  m☉) primarily fuse hydrogen atoms together in a series of stages to form helium, a sequence called the  proton–proton chain. above this mass, in the upper main sequence, the nuclear fusion process mainly uses atoms of  carbon,  nitrogen  and  oxygen  as intermediaries in the  cno cycle  that produces helium from hydrogen atoms. main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. below this mass, stars have cores that are entirely radiative with convective zones near the surface. with decreasing stellar mass, the proportion of the star forming a convective envelope steadily increases. main-sequence stars below 0.4  m☉undergo convection throughout their mass. when core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen.in general, the more massive a star is, the shorter its lifespan on the main sequence. after the hydrogen fuel at the core has been consumed, the star  evolves  away from the main sequence on the hr diagram, into a  supergiant,  red giant, or directly to a  white dwarf.

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