Write down an expression for the velocity of a satellite of mass m in a circular orbit of radius R around the Earth. Take the mass of the Earth as M and use 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. Display response

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.

A pirate has buried his treasure on an island with five trees located at the points (30.0 m, 20.0 m), (60.0 m, 80.0 m), (10.0 m, 10.0 m), (40.0 m, 30.0 m), and (70.0 m, 60.0 m), all measured relative to some origin, as shown in Figure P1.69. His ships log instructs you to start at tree A and move toward tree B, but to cover only one-half the distance between A and B. Then move toward tree C, covering one-third the distance between your current location and C. Next move toward tree D, covering one-fourth the distance between where you are and D. Finally move toward tree E, covering one-fifth the distance between you and E, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as A, B, C, D, and E as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to B (30 m, 20 m), A (60 m, 80 m), E (10 m, 10 m), C (40 m, 30 m), and D (70 m, 60 m)? State reasoning to show that the answer does not depend on the order in which the trees are labeled. Figure 1.69

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?

Two points in a plane have polar coordinates P1(2.500m,/6) and P2(3.800m,2/3). Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.

Figure 1.19 shows two vectors lying in the xy-plane. Determine the signs of the x- and y-components of A, B, and A+B.

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 .

Let u = 8i + 2j and v = 8i − j. (a) Find the angle between u and v. (Round your answer to one decimal place.) (b) Find the component of u along v. (c) Find projvu.

Problem 1: Suppose A = (-3.03 m)i + (4.35 m)j, B = (2.71 m)i + (-4.29 m)j + (2.99 m)k, and D = (-2.48 m)i + (-4.5 m)j. Part (a) What is the angle, in degrees, between D and A? Numeric : A numeric value is expected and not an expression. Part (b) What is the angle, in degrees, between D and B? Numeric : A numeric value is expected and not an expression.

For vectors B = -2î – 8j and A = -6î – 4j, calculate the following. (a) Á + B (Express your answer in vector form.) A + B = -8i – 12j Enter the magnitude and direction (in degrees counterclockwise from the +x-axis) for A + B. magnitude 14.42 direction 56 ° counterclockwise from the +x-axis (b) Á - B (Express your answer in vector form.) A - B = -4i + 4j Enter the magnitude and direction (in degrees counterclockwise from the +x-axis) for A – B. magnitude 5.66 direction -45 ° counterclockwise from the +x-axis