During the siege of the Alamo, both Mexicans and Texans fired at each other with cannons. Along the Alamo walls were 18-pound cannons. These cannons weighed an average of 1270.059 kg. They fired an 18-pound (8.1647 kg) cannonballs. If the velocity of the cannonball peaks at 248 m/s, what was the recoil velocity of the cannon?

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

(a) assume the equation x = at^3 + bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. determine the dimensions of the constants a and b. (use the following as necessary: l and t, where l is the unit of length and t is the unit of time.) (b) determine the dimensions of the derivative dx/dt = 3at^2 + b. (use the following as necessary: l and t, where l is the unit of length and t is the unit of time.)

The radius of a sphere is increasing at a rate of 9 cm/ sec. find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.