An uncrewed mission to the nearest star, Proxima Centauri, is launched from the Earth's surface as a projectile with an initial speed of 43.1 km/s, just enough for the spacecraft to escape the Earth's gravity and leave the solar system. Ignoring air resistance and the Earth's rotation, what is the speed of the spacecraft when it is more than halfway to the star? Assume we are ignoring the effect of the Sun on the spacecraft.

A particle starts from the origin at t = 0 with a velocity of (16 - 12) m/s and moves in the xy plane with a constant acceleration of  = (3.0 - 6.0) m/s2. What is the speed of the particle at t = 2s?

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93×1022kg8.93×1022kg and a radius of 1821 km. How high would this material go on earth if it were ejected with the same speed as on Io? (RE = 6370 km, mE=5.96×1024kg)

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.

Suppose we race a disk and a hollow spherical shell, like a basketball. The spherical shell has a mass M and a radius R; the disk has a mass 2M and a radius 2 R. If the two objects are release at rest, and the height of the ramp is h=.81 m, find the speed of the disk and the spherical shell when they reach the bottom of the ramp

Some of the most spectacular objects in the universe are fairly small: neutron stars are spheres with a diameter that's about 10km (they are remnants of old stars). Their spectacular nature is indicated by the fact that they are about as massive as the sun, i.e. 2 x 1030kg. What is the gravitational acceleration on the surface of such an object? Express your result as a multiple of g.

A record is played by spinning the record so that a circular groove in the vinyl slides under the stylus.  Suppose that a record turns at a rate of 33 1/3 rev. min, the groove being played is at a radius of 10.0 cm and the bumps in the grove are uniformly separated by 1.75 mm.  At what rate (hits per second) do the bumps hit the stylus?  Enter units as hertz (Hz).  You will need v=rω ω is in rad/s and 1rev = 2 pi rad.

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

After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 46.0 cm. The explorer finds that the pendulum completes 98.0 full swing cycles in a time of 145 s. What is the magnitude of the gravitational acceleration on this planet? Express your answer in meters per second per secondgPlanet=(?)m/s^2

Two planets of mass 3 x 1025kg are release from rest 2.0 x 1012 m apart.  What is the speed of the planets when they are half this distance apart?