A roller coaster has a speed of 2.0 m/s at the top of a 60.0-m tall hill as shown below. If friction is negligible, what is the speed of the roller coaster at the top of the loop that has a radius of 20.0 m?
The charged sphere shown below has a net positive charge of 1 nC and is very massive compared to a single proton.
The path of a pole vaulter is measured using video analysis. The y vs. x graph (which shows the path) of the pole vaulter is shown below.
The mass of the vaulter is 70.0 kg. He plants the pole when he is at the location , and the pole starts to bend. With the pole mostly bent, the vaulter starts his mostly upward trajectory at the location . The average force by the pole on the pole vaulter during this interval is . What is the total work done on the pole vaulter during this interval? State any assumptions that you make.
The piece of exercise equipment shown below includes two handles that are on each end of two parallel springs, each with a stiffness 500 N/m. To build strength, a person holds the two handles and pulls them apart with equal magnitude forces applied in opposite directions to the handles, thus stretching the springs. Neglect the mass of the springs and assume that they are pulled horizontally.
A free neutron (mass ) is highly unstable, meaning that it won't exist for very long before it decays. When it decays, it decays into a proton (mass , an electron (mass , and an antineutrino.
In one type of fusion reaction, 6 hydrogen nuclei (each nuclei is one proton) eventually, after many steps, form 1 atom of helium (2 neutrons and 2 protons) and two atoms of hydrogen. (Read more about the proton-proton cycle to learn about the process.) The mass of hydrogen is . The mass of helium is . What is the change in the rest energy of the system during this reaction?
In a 3-D computer game, a 5000-kg space probe in space (far from any significant interactions with planets or stars) has four thrusters that fire in the +x, –x, +y, and –y directions, respectively. As the space probe moves from to , there are two thrusters simultaneously firing with forces and respectively. If the initial speed of the space probe is 200 m/s, what is its final speed? State any assumptions that you make.
To send a probe from earth to another outer planet, it is most efficient (i.e. conserves the most fuel) to put the probe into an orbit about Sun so that when it is nearest Sun (perihelion), its path is tangent to the earth's orbit, and when it is furthest from Sun (aphelion), its path is tangent to the outer planet's orbit. Besides the fuel necessary to put the probe into orbit (i.e. escape from Earth) and to make it orbit the outer planet once it gets there, no fuel is necessary during travel from Earth to the outer planet. For this reason, the probe's orbit in this case is called a "least-energy" orbit.
Suppose the probe travels from Earth to Jupiter as shown below. The radius of the Earth's nearly circular orbit is , and the radius of Jupiter's nearly circular orbit is .
If the probe's speed at aphelion should be the same as the speed of Jupiter ( ), what should be the probe's speed at perihelion, when it leaves Earth with its thrusters turned off? Neglect interactions of the probe with Earth and Jupiter, and assume that the probe's motion is completely determined by its interaction with Sun.
The Hubble Space Telescope (HST or "the Hubble") has a mass of about 11,100 kg and is in a nearly circular orbit of altitude 560 km.
Moon has a mass of and a radius of . Moon has no atmosphere, so there is no air resistance. Suppose that a manned lunar rocket on the surface of Moon must launch so that when it is very far from Moon, it has a speed of 1500 m/s. What is the required launch speed? (Treat the rocket as a particle with constant mass.)