To escape Earth’s gravitational field, a rocket must be launched with an initial velocity called the escape velocity. A rocket launched from the surface of Earthhas velocity v (in miles per second) given by v = √2GM/r + v_0^2 - 2GM/R ≈ √192000/r + v_0^2 - 48 where v_0 is the initial velocity, r is the distance from the rocket to the center of Earth, G is the gravitational constant, M is the mass of Earth, and R is the radius of Earth (approximately 4000 miles). (c) A rocket launched from the surface of a planet has velocity v (in miles per second) given by v = √10600/r + v_0^2 - 6.99. Find the escape velocity for this planet. Is the mass of this planet larger or smaller than that of Earth? (Assume that the mean density of this planet is the same as that of Earth.)

Determine all values of the constant a such that the following function is continuous for all real numbers. f(x) = {ax/tan x, x ≥ 0 {a²-2, x < 0

Two neutron stars are separated by a distance of $1.0 \times 10^{10} \text{~m}$. They each have a mass of $1.0 \times 10^{30} \text{~kg}$ and a radius of $1.0 \times 10^5 \text{~m}$. They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving when their separation has decreased to one-half its initial value?

A satellite that always looks down at the same spot on the Earth’s surface is called a geosynchronous satellite. a. Find the distance of this satellite from the surface of the Earth. b. Could the satellite be looking down at any point on the surface of the Earth?

Three protons are initially very far apart. Calculate the work that must be done in order to bring these protons to the vertices of an equilateral triangle of side

$$5.0 \times 10 ^ { - 16 } \mathrm { m }$$

.

Two equal but opposite charges are placed on the $x$-axis as shown. The positive charge is placed to the left of the origin, and the negative charge is placed to the right. Copy the figure on a piece of paper. Sketch the direction of the net electric field at points $A$ and $B$.

Zero, a hypothetical planet, has a mass of $5.0 \times 10^23 \text{~kg}$, a radius of $3.0 \times 10^6 {~m}$, and no atmosphere. A $10 \text{~kg}$ space probe is to be launched vertically from its surface. If the probe is launched with an initial energy of $5.0\times 10^7 \text{~Joule}$, what will be its kinetic energy when it is $4.0\times 10^6 \text{~m}$ from the center of Zero?

Find all values of c such that the domain of f(x) = x+3 / x²+3cx+6 is the set of all real numbers.

Two long parallel plates are separated by a distance of 15.0 cm. The bottom plate is kept at a potential of −250 V and the top at

$$+ 250 \mathrm { V }$$

. A charge of

$$- 2.00 \mu \mathrm { C }$$

is placed at a point 3.00 cm from the bottom plate. a. Find the electric potential energy of the charge. The charge is then moved vertically up to a point 3.00 cm from the top plate. b. What is the electrical potential energy of the charge now? c. How much work was done on the charge?

The Hubble Space Telescope is in orbit around the Earth at a height of 560km above the Earth’s surface. Take the radius and mass of the Earth to be

$$6.4 \times 10 ^ { 6 } \mathrm { m } \text { and } 6.0 \times 10 ^ { 24 } \mathrm { kg }$$

, respectively a. Calculate Hubble’s speed. b. In a servicing mission, a Space Shuttle spotted the Hubble telescope a distance of 10km ahead. Estimate how long it took the Shuttle to catch up with Hubble, assuming that the Shuttle was moving in a circular orbit just 500m below Hubble’s orbit.

If c is the total cost in dollars to produce q units of a product, then the average cost per unit for an output of q units is given by $\bar{c}=c / q.$ Thus, if the total cost equation is c=5000+6q, then $\bar{c}=\frac{5000}{q}+6.$ For example, the total cost of an output of 5 units is $5030, and the average cost per unit at this level of production is$1006. By finding $\lim _{q \rightarrow \infty} \bar{c},$ show that the average cost approaches a level of stability if the producer continually increases output. What is the limiting value of the average cost? Sketch the graph of the average-cost function.

The masses and coordinates of three spheres are as follows: 20 kg, x = 0.50 m, y = 1.0 m; 40 kg, x = -1.0 m, y = -1.0 m; 60 kg, x = 0 m, y = -0.50 m. What is the magnitude of the gravitational force on a 20 kg sphere located at the origin due to these three spheres?

A spacecraft of mass 30 000 kg leaves the Earth on its way to the Moon and lands on the Moon. Plot the spacecraft’s potential energy as a function of its distance from the Earth’s centre. (The Earth-Moon distance is

$$3.8 \times 10 ^ { 8 } \mathrm { m }$$

. Take the mass of the Earth to be

$$5.97 \times 10 ^ { 24 } \mathrm { kg }$$

and the mass of the Moon to be

$$7.35 \times 10 ^ { 22 } \mathrm { kg }$$

.)

A charge q of 10.0 C is placed somewhere in space. What is the work required to bring a charge of 1.0 mC from a point X, 10.0 m from q, to a point Y, 2.0 m from q? Does the answer depend on which path the charge follows?