A spacecraft arrives at Mars’ sphere of influence with a heliocentric velocity of 22 km/s and a heliocentric fight-path angle of 10 deg. When the spacecraft reaches the periapsis of its Mars-centric arrival trajectory, at an altitude of 200 km, it performs a ∆V to circularize its trajectory. What is the required ∆V ?

Given: LST 81.1 deg W, 29.2 deg N, Ω = 150◦,i = 30◦. Find: Time in hours until the next launch window opens and launch azimuth β.

520/5000

State the hypotheses of the causes of why: of each of the following alternatives.

A) automobile (without gasoline and without brake) slows down and stops;
b) a book that slides on the table slows down and stops;
c) a falling chalk increases its speed until it hits the ground;
d) a ball rolling down a slope increases its speed until it reaches the lowest part;
e) The Moon moves around Earth and moves with approximately constant speed.

Part two:

Design an experiment where you check that the acceleration coincides with the direction of the resulting force.

An 60 kg plank, 15 m long, has supports at locations X1=4 m and X2=12 m along the plank. Two additional forces F3=20 N and F4=25 N act at the points X3=2 m and X4=10 m along the plank, making angles of 30 degrees and 60 degrees with respect to the vertical.

a) from x-equilibrium, determine the value of F4,

b) from rotational equilibrium, determine the support forces N1 and N2 acting at the location of the two supports, taking into account the mass of the plank itself,

c) show that your answers are correct by demonstrating that they satisfy equilibrium in y-.(i.e., that the sum of the forces in the y-direction are zero, since the plank is stationary).

(a) What is the escape speed on a spherical asteroid whose radius is 376 km and whose gravitational acceleration at the surface is 0.683 m/s²? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 445 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 592.7 km above the surface?

A parallel-plate capacitor is constructed of two horizontal 17.2-cm-diameter circular plates. A 1.1 g plastic bead with a charge of -8.0 nC is suspended between the two plates by the force of the electric field between them. What is the charge on the positive plate?

n the Group Work, we examine standing waves on a string. In order to calculate the velocity of the wave, what two properties do we need to know?

show the velocity and direction while proving that s(x,t) = Aexp[-(4x+6t)^2] is a wave equation

A uniform hoop of mass M and radius R rolls down an incline without slipping, starting from rest. The angle of inclination of the incline is θ.

a. After moving a distance L along the incline, what is the angular speed ω of the hoop?

b. If the coefficient of static friction between the hoop and the incline is µs = 1/3, what is the greatest possible value of θ such that no slipping occurs between the hoop and the incline?

c. What is the direction of the angular velocity for the rolling hoop?

Using a coil of wire and an oscilloscope, explain how you can detect the presence of the Earth's magnetic field.

Using a coil of wire and an oscilloscope, explain how you could show that a magnet is producing a magnetic field.

A series LRC circuit is driven at a frequency which is much less than the resonant frequency with a sine wave voltage source V0sin(wt). What is the approximate voltage across the capacitor? Briefly explain

A series LRC circuit is driven at resonance with a sine wave voltage source. What is the sum of the voltages across the inductor and the capacitor in an ideal circuit? Briefly explain.

2. For a system of non-interacting, one-dimensional, distinguishable classical particles in a harmonic oscillator potential, V = kx² in contact with a particle reservoir with chemical potential µ and a thermal reservoir at temperature T.

(a) Calculate the grand partition function Z for the system. Note that there is no fixed "volume" for this system.

(b) Obtain N (number of particles) and U as functions of µ and T and show that U satisfies the equipartition theorem

A 3.4 μC charge is placed at the origin of coordinates, and a -2.6 μC charge is placed to the x axis at 5.0 cm . 1)Find the location of the place(s) along the x axis where the electric field due to these two charges is zero. Express your answer(s) using two significant figures. If there is more than one answer, enter each answer separated by a comma. 2)Find the location of the place(s) along the x axis where the electric potential due to these two charges is zero. Express your answer(s) using two significant figures. If there is more than one answer, enter each answer separated by a comma?

- estimate the energy (in joule) produced from radioactivity over the Earth's lifetime.

- Estimate the energy density (in joule/kg) of nuclear fission fuels.

- Estimate mass (in kg) of radioactive material is need to power Earth’s interior over its lifetime.

- Estimate the mass of Earth (in kg).

- Estimate the fraction of Earth's mass in radioactive material.

Consider the 1D wave equation d^2u/dt^2 = c^2( d^2u/dx^2) with the following boundary conditions: u(0, t) = ux (L, t) = 0 . (a) Use separation of variables technique to calculate the eigenvalues, eigenfunctions and general solution. (b) Now, assume L = π and c = 1. With initial conditions u(x, 0) = 0 and ut(x, 0) = 1, calculate the solution for u(x, t). (c) With initial conditions u(x, 0) = sin(x/2) and ut(x, 0) = 2 sin(x/2) − 3 sin(5x/2) calculate the solution for u(x, t).