Let A = 2 1 1

1 2 1

1 1 2

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

Charlie kicks a soccer ball up a small incline. On the way up, ball’s acceleration has magnitude |a| = 0.45 m/s2 and is directed in downhill direction. Charlie kicks the ball at the bottom of the incline and then immediately start to walk up the incline with constant speed. Charlie performs twi different trials. a) In the first trial, Charlie kicks the ball with initial speed v0 = 3.4 m/s. Charlie is 2.3-m behind the ball when the ball is at the highest point. What is the speed vC of Charlie? b) Charlie performs the second trial. He kicks the ball with unknown speed v 0 0 but walks with the same speed vC as in the first trial. Charlie is now 0.8 m behind the ball when the ball is at the highest point. What is the initial speed v 0 0 of the ball at the bottom of the hill? (Hint: You need to set-up a quadratic equation for v 0 0 ).

Two identical blocks of mass M = 2.60 kg each are initially at rest on a smooth, horizontal table. A bullet of very small mass m = 20 g (m << M) is fired at a high speed v. = 120 m/s towards the first block. It quickly exits the first block at a reduced speed of 0.40 v, then strikes the second block, quickly getting embedded inside of it. All the motion happens on the x-axis.

(a) find the speeds of the two blocks after their encounters with the bullet.

(b) Now the first block catches up with the second one and collides with it. They got stuck together afterward and move forward. Find their common speed V after the collision.

(c) The two blocks now hit a light spring of spring constant k = 35 N/m mounted on the wall. How far is the spring compressed before the blocks reach a momentary stop?

AFM Co. has a market value-based D/V ratio of 1/3. The expected return on the company’s unlevered equity is 20%, and the pretax cost of debt is 10%. Sales for the company are expected to remain stable indefinitely at $25 million. Costs amount to 60% of sales. The corporate tax rate is 30%, and the company distributes all its earnings as dividends at the end of each year. The company’s debt policy is to maintain a constant market value-based D/V ratio.

(a) If the company were all equity financed, how much would it be worth?

(b) What is the expected rate of return on the firm’s levered equity?

(c) First, use the after-tax WACC approach to calculate the value of the entire company (V). Then compute the value of the company’s equity (E) and the value of the company’s debt (D).

(d) Use the APV approach to compute the value of the company. You will need to use some of the answers from part (c).

Let u(t) and w(t) be the horizontal and vertical components, respectively, of the velocity of a batted baseball:

??=−?? and ??=−?−?? ?? ??

where ? is the coefficient of air resistance and g is the acceleration of gravity.

1. Determine u(t) and w(t) assuming the following initial conditions: ?(0) = ? cos ? and ?(0) = ? sin ?,where V is the initial speed of the ball, and A is its initial angle of elevation.

2. Let x(t) and y(t) be the horizontal and vertical coordinates, respectively, of the ball at time t. If?(0) = 0 and ?(0) = h, find x(t) and y(t) at any time t.

3. Plotthetrajectoryoftheballfor?=1/5,V=125ft/s,h=3ft,A=0.5rad,andg=32ft/s2.

4. Suppose that the outfield wall is at a distance L and has height H. Assuming that ? = 1/5, h = 3, L =

   300 ft, and H = 10 ft, find the minimum initial velocity V and the optimal angle of elevation A for which the ball will clear the wall.

A skydiver, weighing 180 lb (including equipment) falls vertically downward from an altitude of 4000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.75|v| when the parachute is closed and is of magnitude 10|v| when the parachute is open, where the velocity v is measured in ft/s. (A computer algebra system is recommended. Use g = 32 ft/s2 for the acceleration due to gravity. Round your answers to two decimal places.)

(a) Find the speed of the skydiver when the parachute opens.

(b) Find the distance fallen before the parachute opens.

(c) What is the limiting velocity v_L after the parachute opens?

(d) Determine how long the sky diver is in the air after the parachute opens.

e) Plot the graph of velocity versus time from the beginning of the fall until the skydiver reaches the ground.

A 72.0 kg swimmer jumps into the old swimming hole from a tree limb that is 3.95 m above the water.

Use energy conservation to find his speed just as he hits the water if he just holds his nose and drops in.

Express your answer to three significant figures.

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Part B

Use energy conservation to find his speed just as he hits the water if he bravely jumps straight up (but just beyond the board!) at 2.60 m/s .

Express your answer to three significant figures.

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Part C

Use energy conservation to find his speed just as he hits the water if he manages to jump downward at 2.60 m/s .

Express your answer to three significant figures.

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Write a function in Matlab that takes as input the number n and a symmetric tridiagonal matrix given as two vectors: n×1 vector v representing the main diagonal and (n−1)×1 vector w representing the upper diagonal. Have this function output the Cholesky factor of the matrix as a vector for the main diagonal and a vector for the upper diagonal and output the number of flops and, separately, the number of square roots used as well. Use only basic programming.

(a) Write out or print out your function.

(b) Run the case with v =2*ones(10,1), w = -ones(9,1) and write out or print out all your results.

(c) Rune the case with v =2*ones(100,1), w =-ones(99,1) and write out or print out your results just for the number of flops and square roots used. How many times more flops are used than in the previous case?

c = 3.0 x 10⁸ m/s;        b = v/c;           g (gamma) = 1/sqrt[1 – b²];    u = (u’ + v) / (1 + u’v/c²);

E_ph = hf = hc/l;           P = esAT⁴;     s = 5.67 x 10^-8 Watts / m² / K⁴

1. Relativistic Factor. Calculate the value of the relativistic factor g for an object traveling at v = 0.999c.

2. E = mc². If precisely 1.00 kilogram of mass could be converted completely into energy, for how long could this amount of energy keep a 100-Watt light bulb shining? (Assume the bulb can operate for a very long time.)

3. Velocity Addition. A relativistic craft traveling at 0.6c w/r/t to the Earth launches a probe, which travels at a velocity of 0.8c w/r/t the craft. What is the velocity of the probe w/r/t the Earth? (It’s not 1.4c.)

1. Star with Newton's third law: dp/dt = ΣF where p is the momentum. In space, the sum of the external forces ΣF = 0. For a rocket in space, the mass and velocity change with time as the rocket expends its fuel. Show from Newton's third law that ΔV = v ln(m/M) where m is the initial mass of the rocket, M is the final mass of the rocket after the fuel is expended, v is the rocket's exhaust velocity and ΔV is the change in the velocity for the rocket

2. Now, the rocket needs to get to escape velocity to get to the moon. Use conservation of energy to derive an expression for the escape velocity. If we start at 200km above the surface of earth, calculate the ΔV we need to escape from Earth and get to the moon.

3. Finally, use the earlier Rocket Equation calculate the fraction of the initial rocket's mass that is dedicated to fuel to make the journey from the Earth to the Moon. In this case, v is 4500 m/s.