Suppose it is possible to dig a smooth tunnel through the earth from a city at A to a city at B as shown.(Figure 1) By the theory of gravitation, any vehicle C of mass m placed within the tunnel would be subjected to a gravitational force which is always directed toward the center of the earth D. This force F has a magnitude that is directly proportional to its distance r from the earth's center. Hence, if the vehicle has a weight of W=mg when it is located on the earth's surface, then at an arbitrary location r the magnitude of force F is F=(mg/R)r, where R=6328 km, the radius of the earth. Part A) If the vehicle is released from rest when it is at B, x=s=2 Mm, determine the time needed for it to reach A. Neglect the effect of the earth's rotation in the calculation and assume the earth has a constant density. Hint: Write the equation of motion in the x direction, noting that rcosθ=x. Integrate, using the kinematic relation vdv=adx, then integrate the result using v=dx/dt. Part B) Determine the maximum velocity the vehicle attains

Suppose it is possible to dig a smooth tunnel through the earth from a city at A to a city at B as shown.(Figure 1) By the theory of gravitation, any vehicle C of mass m placed within the tunnel would be subjected to a gravitational force which is always directed toward the center of the earth D. This force F has a magnitude that is directly proportional to its distance r from the earth's center. Hence, if the vehicle has a weight of W=mg when it is located on the earth's surface, then at an arbitrary location r the magnitude of force F is F=(mg/R)r, where R=6328 km, the radius of the earth. Part A) If the vehicle is released from rest when it is at B, x=s=2 Mm, determine the time needed for it to reach A. Neglect the effect of the earth's rotation in the calculation and assume the earth has a constant density. Hint: Write the equation of motion in the x direction, noting that rcosθ=x. Integrate, using the kinematic relation vdv=adx, then integrate the result using v=dx/dt. Part B) Determine the maximum velocity the vehicle attains

Similar questions

(a) What is the escape speed on a spherical asteroid whose radius is 803 km and whose gravitational acceleration at the surface is 1.75 m/s2? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1330 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 1811 km above the surface?
(a) Evaluate the gravitational potential energy (in J) between two 4.00 kg spherical steel balls separated by a center-to-center distance of 19.0 cm. (b) Assuming that they are both initially at rest relative to each other in deep space, use conservation of energy to find how fast (in m/s) will they each be traveling upon impact. Each sphere has a radius of 5.50 cm. m/s
On the ground level, the weight of a satellite is W N. The satellite is launched and stays in an orbit that is R/47 above the ground level, where R is radius of the Earth. If the gravitational force on the satellite is xW at this location, determine x.
One of your summer lunar space camp activities is to launch a 1090 kg rocket from the surface of the Moon. You are a serious space camper and you launch a serious rocket: it reaches an altitude of 227 km. What gain in gravitational potential energy does the launch accomplish? The mass and radius of the Moon are 7.36×10^22 kg and 1740 km, respectively.
In 2000 , NASA placed a satellite in orbit around an asteroid. consider a spherical asteroid with a mass of 1.50 * 10^16 kg and a radius of 10.0 km what is the speed of a satellitie orbiting 5.90 km above the surface ? express with appropriate units what is the escape speed from the asteroid (express with appropriate units)
One of your summer lunar space camp activities is to launch a 1210-kg rocket from the surface of the Moon. You are a serious space camper and you launch a serious rocket: it reaches an altitude of 229 km. What gain in gravitational potential energy does the launch accomplish? The mass and radius of the Moon are 7.36 × 1022 kg and 1740 km, respectively.
A satellite moves in a circular orbit around Earth at a speed of 4381 m/s. (a) Determine the satellite's altitude above the surface of Earth. Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. m (b) Determine the period of the satellite's orbit. Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. h Need Help? Read It
Two concentric spherical shells with uniformly distributed masses M, and M, are situated as shown in the figure below. a M1 M2 Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at the following radial distances. (Use any variable or symbol stated above along with the following as necessary: a, b, c, and G for the gravitational constant.) (a) r = a F = (b) r = b F = (c) r = c F =
Two concentric spherical shells with uniformly distributed masses M₁ and M₂ are situated as shown in the figure. Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at each of the radial distances shown in the figure. Fa NOTE: Give your answer in terms of the variables given and G when applicable (a) What is the magnitude of the net gravitational force if the particle is located outside both shells with a radial distance a? F = M₁ (b) What is the magnitude of the net gravitational force if the particle is located between the two shells with a radial distance b? Fc M₂ ****** - a (c) What is the magnitude of the net gravitational force if the particle is located inside both shells with a radial distance c? =
a) What is the total energy needed to place a 2.0 x 103 kg satellite into circular Earth orbit at an altitude of 5.0 x 102 km? b) How much additional energy would have to be supplied to the satellite once it was in orbit, to allow it to escape from Earth’s gravitational field?
A 2660-kg spacecraft is in a circular orbit 1540 km above the surface of Mars. How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 4500 km above the surface? Express your answer to three significant figures.