I keep getting this wrong despite working it out many times. Can I please get some insight on the right approach? A team of astronauts is on a mission to land on and explore a large asteroid. In addition to collecting samples and performing experiments, one of their tasks is to demonstrate the concept of the escape speed by throwing rocks straight up at various initial speeds. With what minimum initial speed ?esc will the rocks need to be thrown in order for them never to "fall" back to the asteroid? Assume that the asteroid is approximately spherical, with an average density ?=3.32×106 g/m3 and volume ?=2.40×1012 m3 . Recall that the universal gravitational constant is ?=6.67×10−11 N·m2/kg2 . Answer in m/s

Ex. 3: Consider the below forces, which depend on the position of the particle. Imagine a mass (m) which is subject to these forces. Suppose it is observed that the mass is not moving at x = 0. 1. F(x) = F + c. x II. F(x) = F sin(cx) a) Compute the velocity as a function of position of the mass. b) Determine the kinetic energy of the mass. III. F(x) = F ex c) Find the potential energy of the particle as a function of position d) For the force in part (1), find the amount of time it takes for the mass to traverse a 1m distance.

A.) i. Define the plane polar unit vectors ŕ and 0 and show that they are orthonormal. Determine dr dê and de de ii. Hence show that for the vector r(t) = r(t)ŕ dr · = rî +rðê, d²r and = - dt dt² († − rė²)ŕ + (2ŕė +rë)ê . iii. If a mass m moves in the plane along r(t) = a and (t) = 5t determine the shape of the motion and find its period. B) Show that the motion of an object under the universal gravitational force μη Fg is governed by the radial and angular equations of motion and μ = 0, 2r0+ rö = 0. Setting r(t) =R for some constant R, solve these equations of motion and determine the period of the motion. C) Consider the following non-linear differential equation *+5= x +4. Find the stationary solutions. Expand around each of the stationary so- lutions that you have found, in order to determine if they correspond to stable or unstable equilibria. D) Show that 1 5 E = 2 - 4x is a conserved quantity. Hence draw the phase-space diagram and de- termine for which values of E oscilations…

A 2000 kg lunar lander is in orbit 30 km above the surface of the moon. It needs to move out to a 500 km -high orbit in order to link up with the mother ship that will take the astronauts home. Part A How much work must the thrusters do? Express your answer with the appropriate units. μА Value Units ?

In a futuristic scenario, you are assigned the mission of making an enemy satellite that is in a circular orbit around Earth inoperative. You know you cannot destroy the satellite, as it is well protected against attack, but you can try to knock it out of its orbit so it will fly away and never return. What is the minimum amount of work ?W applied to the satellite that is required to accomplish that? The satellite's mass and altitude are 969 kg969 kg and 241 km.241 km. Earth's mass and radius are 5.98×1024 kg5.98×1024 kg and 6370 km.

What are the two postulates of special relativity? (Choose two answers.) A In any closed system, the total momentum will not change. B F=ma (c) The entropy of an isolated system will always increase over time. The speed of light is the same regardless of the speed of the observer or the speed of the source. Light travels to us from the hands of the clock. If we could move away from the clock at the speed of light, time will appear to stand still. If you measure both the position of a particle and its momentum, then the product of the F uncertainties of these two measurements must always be at least half of the reduced Planck's constant. For all possible paths between two points, the actual trajectory is the one that results in the action being minimized.

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip.a. How fast must the rocket travel relative to earth?b. How much energy is needed to accelerate the rocket to this speed?c. How many times larger is this energy than the total energy used by the United States in a typical year, roughly 1.0 x 1020 J?

I just need to know what is the Energy Conservation Equation.

A Fall on the Moon. You and your team are stationed at an earth-based mission control facility and have been tasked with assisting astronauts via radio communication as they exit a spacecraft that has recently landed on the moon. The astronauts intended to explore the lunar surface but, unfortunately, they hit a snag even before they opened the hatch. Their craft has landed right on the edge of a 15-ft cliff, so that there is no way off the lander except to drop that distance from the bottom rung of a ladder on the side of the ship. You look up the structural parameters of the space suits, the gear they are wearing, and the physical limitations of the crew, and find that, on earth, the maximum drop they can safely endure is 4.0 ft. (a) Is it safe for them to make the 15-ft drop on the moon? (b) What is the maximum drop they can safely endure on the moon? Note that the acceleration due to gravity on the moon is one-sixth as large as that on earth.

OSCILLATORY MOTION 9. In Example 2 of Chapter 7 we found that the gravitational force on a mass m inside a spherical earth of uniform density p is F = kr, directed toward the cen- ter, with k = TGpm. (a) For a mass dropped from the surface of the earth down a tunnel straight through the center of the earth, find the time T, to pass entirely through the earth and back again. (b) Compute the speed of the mass m as it passes through the center of the earth. (c) Compare your result for T, with the time 72 required to complete a circular orbit around the globe just above the earth's surface.