Physics 1
EXAM 2
Prof. Michael Sobel

Answers will be found at the following link.

Show all work. Part credit will be given.

The gravitational constant, G = 6.67 X 10-11 N-m2/kg2.
Mass of the earth, ME = 5.98 X 1024 kg.
The product GME = 3.99 X 1014 N-m2/kg.
Radius of earth, RE = 6.4 X 106 m.

ANSWER FOUR OF THE SIX QUESTIONS.

1. A neutron moving at 8.0 X 107 m/s makes a head-on elastic collision with a helium nucleus. (Mass of the helium = 4 X mass of the neutron.)
     a) Find the speed of the neutron after the collision.
     b) Which direction is the neutron moving (forward or back)?
     c) Let f = (final kinetic energy of the neutron)/(initial kinetic energy of the neutron). Show that f = 0.36.
     d) Suppose that for many elastic collisions, not necessarily head-on, the average value of f = 0.2. How many collisions are needed, on the average, to reduce the neutron’s kinetic energy by a factor of 106?

2. In the following, neglect air resistance. A rocket is launched vertically from the earth’s surface with speed v0 = 2.5 X 103 m/s. When it reaches a height h above the earth’s surface, its distance from the center of the earth is r = RE + h. Its speed is v.
     a) Write the conservation of energy equation, which relates r and v.
     b) The rocket reaches a certain maximum value of r, and then turns around and falls down to earth. Write the equation that determines r.
     c) Find this value of r and find h.

3. A planet is in a circular orbit around the sun, with a radius,r.
     a) Write an equation that relates the speed v to r (and to G and MS, the mass of the sun). Solve for v in terms of the other quantities.
     b) Write an equation that relates T (the period of the planet’s orbit) to r and v.
     c) Solve for T, and show that T is proportional to r3/2.

4. A mass = 0.5 kg is attached to a spring with k = 2.0 N/m. The mass slides on a horizontal frictionless surface. It is in equilibrium at x = 0. The mass is now moved to the right a distance = 0.6 m and then released from rest.
     a) Find the total energy of the system.
     b) Where (what value of x) does the mass have its maximum velocity? Explain briefly in words.
     c) Find the maximum velocity.
     d) Find the speed of the mass at x = 0.3 m.

5. A cylinder of height h and cross-sectional area A is immersed in a fluid of density r. Starting with the equation for the pressure at a given depth in a fluid,
     a) Find an expression for the force on the bottom of the cylinder due to the fluid. What is its direction?
     b) Find an expression for the force on the top of the cylinder due to the fluid. What is its direction?
     c) Find an expression for the net upward force on the cylinder – in terms of r, g, and the volume of the cylinder. (This is the buoyant force, B.)

6.
     a) Write (don't derive) the kinetic theory equation for the pressure of an ideal gas – in terms of the mass and velocity of molecules.
     b) Suppose a gas is kept in a container of volume = 0.012 m3 at pressure = 1.1 X 105 Pa. If the mass of the gas is 0.015 kg, find the average molecular velocity.
     c) Why is it that we cannot deduce the mass of a gas molecule from the equation in part (a)?
     d) Suppose the temperature (in Kelvins) is doubled (the gas being kept in the same container). Use your answer to part (b), to determine what the average molecular velocity will be? (If you don’t have an answer for part (b), assume the answer to (b) is 400 m/s.)