In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy. The kinetic energy K of an object of mass m and velocity v is given by K = }mv². Suppose an object of mass m, moving in a straight line, is acted on by a force F= F(s) that depends on its position s. According to Newton's Second Law dv m dt F(s) = ma where a and v denote the acceleration and velocity of the object. (a) Show that the work done in moving the object from a position so to a position s is equal to the change in the object's kinetic energy; that is, show that - S" F(s) ds = }mv? – }mv3 where vo = v(so) and vi = v(s1) are the velocities of the object at the positions so and s1. Hint: By the Chain Rule, dv dv ds dv m dt ds dt mv ds (b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?

A luggage handler pulls a 20.0 kg suitcase up a ramp inclined at 34.0 ∘∘ above the horizontal by a force F⃗ → of magnitude 163 NN that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is μk = 0.300. The suitcase travels 3.70 mm along the ramp.  Calculate the work done on the suitcase by F⃗ →. Calculate the work done on the suitcase by the gravitational force.   Calculate the work done on the suitcase by the normal force.   Calculate the work done on the suitcase by the friction force.   Calculate the total work done on the suitcase.   If the speed of the suitcase is zero at the bottom of the ramp, what is its speed after it has traveled 3.70 mm along the ramp?

An object of mass m = 3 kg moves in the x direction with the potential energy U (x) shown. There are no other forces acting. On the axes provided below the graph of U (x), sketch the force F(x) that the mass experiences. If the object is released from rest at x = -2 where U = 16J, what is its velocity at x = O where U = -8J? %3D U(x) (J) |16 х (m) F(x) (N) х (m)