A student club is designing a trebuchet for launching a pumpkin into projectile motion. Based on an analysis of their design, they predict that the trajectory of the launched pumpkin will be parabolic and described by the equation y(x) = ax² + bx where a = −8.0 × 10−³ m−¹, b = 1.0 (unitless), x is the horizontal position along the pumpkin trajectory and y is the vertical position along the trajectory. The students decide to continue their analysis to predict at what position the pumpkin will reach its maximum height and the value of the maximum height. What is the derivative of the vertical position of the pumpkin trajectory with respect to its horizontal position? ▸ View Available Hint(s) dy dx = 2ax + b dy dx dy dx dy dx - 0 = 2ax ax

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?

Two vectors are given by a  =  1.5î−4.0ĵ and b  =  −11.9î + 8.1ĵ. What is the magnitude of b? i already found out  magnitude of b is 1.44 but how to calculate the angle between vector b and the positive x-axis?

The vector product of vectors A → and B → has magnitude 12.0 m2m2 and is in the +z+-direction. Vector A → has magnitude 4.0 mm and is in the −x− -direction. Vector B → has no x-component. What is the directional angle θ of vector B → in the xy-plane measured counterclockwise from the +x+ -direction?

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.

What is the angle between the two vectors A = 2i + 3j and B = i- 2j is * A 0° 120° 90° 60°

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

Consider the following vectors: A   =  1.2i + 4.9j + 3.7k   B   =  -8.5i + 7.5j + 8.3k Part (a)  What is the x-component of the vector V = 3(A × 2B)?  Part (b)  What is the y-component of the vector V = 3(A × 2B)?    Part (c)  What is the z-component of the vector V = 3(A × 2B)?    Part (d)  What is the magnitude of the vector V?

On a certain mountain, the elevation z above a point (x,y) in an xy-plane at sea level is z = 2000 −0.02x2−0.04y2, where x, y, and z are in meters. The positive x-axis points east, and the positive y-axis north. Stephie is at the point (20,5,1991). (a) If Stephie uses a compass reading to walk northeast, will she ascend or descend? At what rate?

Refer to the figure. a = 4, b = 3, and c = 5, what are (a) the magnitude and the direction of å x b, (b) the magnitude and the direction of å x č, and (c) the magnitude and the direction of Bx ở ? The +z axis is out of the plane.