Beth exerts 14 N of force to propel a 4.5 kg bowling ball down the lane. Describe how the ball will travel

(1) While Beth continues exerting the force of 14N, the ball will experience an acceleration of (Force) / (Mass) = 14N/4.5kg=3.1 m/s^2.

After the ball leaves Beth's hand (and the acting force will cease), it will have gained a certain amount of kinetic energy. Assuming no energy loss during the maneuver, the amount of kinetic energy will equal the amount of work Beth did:

(Force) x (displacement) = (Kinetic energy)

If you were given the displacement (the length of the trajectory of the ball while in Beth's hand), you could determine the velocity using either the kinetic energy formula, or the appropriate kinematic equation.

(2) After the ball is on the lane, it tends to move with constant velocity. But, due to friction forces of the lane and air resistance, it will decelerate during this stage. Luckily, the lanes are made so as to minimize such friction, so provided Beth gave the ball enough "umpf" the ball will eventually make it to the other end to hit some of the pins

(3) Upon hitting the pins the ball will change its trajectory due to the (mostly elastic) collision (s). If you were given the exact velocity at that time and positions and masses of the pins, you might be able to use the law of conservation of momentum, to calculate the progression of events from there.

It will have a certain kinetic energy at the beginning, but it will be lost by resistant work done by friction. Thus, itsmotion wil be uniformly decelerating.