Write down an expression for the acceleration of a satellite of mass m in a circular orbit of radius R around the Earth. Use M for the mass of the Earth and G for the gravitational constant. Please use appropriate algebraic symbols for multiplication (* for a × b), division (/ for a/b), exponents (a^b for a³), square root (sqrt(a*b/c) for √√(a × b/c)) etc. a= Display response

Let us name three perpendicular directions as right, up, and toward you as you might name them when you are facing a television screen that lies in a vertical plane. Unit vectors for these directions are r, u, and t, respectively. Consider the quantity (3u2t). (i) Is the magnitude of this vector (a) 6, (b) 3, (c) 2, or (d) 0? (ii) Is the direction of this vector (a) down, (b) toward you, (c) up, (d) away from you, or (e) left?

Vector B is 5.0 cm long and vector A is 4.0 cm long. Find the angle between these two vectors when |A+B|=3.0cm and |AB|=3.0cm .

Two points in the Cartesian plane are A(2.00m,4.00m) and B(-3.00m,3.00m) . Find the distance between them and their polar coordinates.

Check Your Understanding Verify that vector v V obtained in Example 2.14 is indeed a unit vector by computing its magnitude. If the convoy in Example 2.8 was moving across a desert flatland—that is, it the third component of its velocity was zero—what is the unit vector of its direction of motion? Which geographic direction does it represent?

Suppose A = (-2.33 m)i + (3.79 m)j, B = (2.87 m)i + (-4.8 m)j + (2.18 m)k, and D = (-2.99 m)i + (-4.24 m)j..  What is the angle, in degrees, between D and A

Schwarzschild radius RS of a black hole is the maximum distance from the black hole’s center at which light cannot escape its gravitational field. The quantity RS (with dimensions of length) is dependent on the mass of the black hole M, the speed of light c, and the gravitational constant G. Based on the dimensions of these four parameters, predict an equation for the Schwarzschild radius. Hint: G has dimensions of [L3/MT2]

Use the definition of scalar product, a b = ab cos 0, and the fact that a ⋅ b = axbx + ayby+ a₂b₂ to calculate ▪ the angle between the two vectors given by a = 3.01 +3.07 + 3.0k and b 7.0î + 8.07 + 9.0k. Number i Units =

A= 2.00î + 6.00ĵ and B d= A+ B Given the vectors 3.00 î – 2.00j, and the vector difference D= A – B. (b) Calculate C and D, in terms of unit vectors. (c) Calculate C and D in terms of polar coordinates, with angles measured with respect to the positive x axis. (a) draw the vector sum

Kepler's third law states that for any object in a gravitational orbit,  P2∝a3P2∝a3  where PP  is the orbital period of the object and aa  is the average distance between the object and what it is orbiting. In our Solar System, the natural units are distances measured in astronomical units (A.U.) and orbital periods measured in years. This can be seen for the Earth-Sun system which has an orbital period P=1P=1  year and an average distance  a=1a=1  AU. Using these natural units in the Solar System, the proportionality becomes an equality, so for our Solar System:  (Pyears)2=(aA.U.)3(Pyears)2=(aA.U.)3 . Using your mathematical prowess, determine what the orbital period in years would be for an asteroid that was discovered orbiting the Sun with an average distance of 25 astronomical units.