Problem 1: The energy E of a particle of mass m moving at speed v is given by: E2 = m2 c4 + p2 c2 (1) p=γmv (2) 1 γ = 1−v2/c2 (3) This means that if something is at rest, it’s energy is mc2. We can define a kinetic energy to be the difference between the total energy of an object given by equation (1) and the rest energy mc2. What would be the kinetic energy of a baseball moving at half the speed of light? Give your answer in Joules but also in tons of TNT equivalent. For context, the worst nuclear weapons are measured in megatons. Do you think we could ever accelerate a baseball to half the speed of light?

Problem 2: As we said in lecture, if you heat something up, it becomes more massive! Imagine you took 1 kg of water and heated it up by 50 deg C. By how much would it’s mass increase because of this heating? You might have to look up the heat capacity of water.

An AC generator supplies an rms voltage of 230 V at 60.0 Hz. It is connected in series with a 0.300 H inductor, a 4.30 μF capacitor and a 256 Ω resistor.

What is the impedance of the circuit?

What is the rms current through the resistor?

What is the average power dissipated in the circuit?

What is the peak current through the resistor?

What is the peak voltage across the inductor?

What is the peak voltage across the capacitor?

The generator frequency is now changed so that the circuit is in resonance.

What is that new (resonance) frequency?

[(Food ^ Drinks => Party) v (Drinks ^ Dance => Party)] => (Party => Drinks)

Convert the left-hand side of this formula (in the brackets) into CNF and then into clauses. Show all steps of deriving the clauses.

1. Convert the right-hand side of this formula (not in the brackets) into CNF and then into clauses. Show all steps of deriving the clauses.

A long, straight wire carrying a current of 3.00 A moves with a constant speed v to the right. A 5-turn circular coil of diameter 1.25 cm, and resistance of 3.25 µΩ, lies stationary in the same plane as the straight wire. At some initial time, the wire is at a distance

d = 19.5 cm

from the center of the coil. 4.55 s later, the wire is at a distance 2d from the center of the coil. What is the magnitude and direction of the average induced current in the coil? Note that while the magnetic field varies over the diameter of the coil, it is very small and we will disregard this variation.

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line
element is defined by:

ds2 = -xdv2 + 2dvdx

i) Assuming that the nature and properties of the spacetime line element still hold,
show that a particle in the negative x-axis can never venture into the positive
x-axis. That is, show that a particle becomes trapped if it travels into the
negative x-axis.
ii) Draw a representative light cone that illustrates how a particle is trapped.

Draw a circuit with a battery of 90 V is connected to four resistors, R1 =10 Ω, R2,= 20 Ω , R3 =30 Ω and R4= 40 Ω, as follows. Resistors R1 and R2 are connected in parallel with each other, resistors R3 and R4 are connected in parallel with each other, and both parallel sets of resistors are connected in series with each other across the battery. (20 Points)

a. Find the equivalent resistor R12, for the partial circuits R1 and R2,

b. Find the equivalent resistor R34, for the partial circuits R3 and R4,

c. Find the equivalent resistor R1234, for the entire circuits with all four resistors R1, R2, R3, and R4

d. Find the current in the circuit with the single equivalent resistor R1234

e. Find the find the current and voltages across each of the partial circuits with resistors R12,

and R34.

e. Find the current and voltage across each of the four resistors R1, R2, R3, and R4

f. Find the power dissipated in each of the resistors R1, R2, R3, and R4

A fluid flow is defined by u = (0.4x2 + 2t) m/s and v = (0.8x + 2y) m/s, where x and y are in meters and t is in seconds.

Part A

Determine the magnitude of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

V=

Part B

Determine the direction of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θV, which the velocity makes with the x axis, measured counterclockwise from the positive x axis.

.

Part C

Determine the magnitude of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

Part D

Determine the direction of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θa, which the acceleration makes with the x axis, measured counterclockwise from the positive x axis.

In Mr. W Fireworkd, Inc. v. Ozuna, the Court of Appeals decided not to enforce a certain provision of the contract between the landowner, Mr. Ozuna, and the lessee, Mr. W Fireworks. Explain what provision of the lease Mr. W Fireworks sought to enforce and why the court refused to do so.

Let the present value from production be equal to V = 100, and this value can move either up (with factor u = 1.4) or down (with factor d = 1/u) per period. Suppose that at t=3 management has the option to invest 130 million in order to double the value of production. The risk free rate is 2%.

What is the expanded present value of this production facility if management has the opportunity to expand at t = 3

A person standing far away walks toward a plane mirror at a speed v, and the speed of the image is 2v relative to the person. Please compute the speed of the image relative to the person if the plane mirror is replaced by (a), a convex spherical mirror? (b), a concave spherical mirror?