Final answer:
The speed of the 40.0 kg wagon after moving up the hill without friction is obtained using the work-energy principle. The work done by the tow rope minus the work done against gravity provides the net work, which is equal to the change in the wagon's kinetic energy. From this, we can calculate the wagon's speed after moving 30.0 m and 80.0 m up the hill.
Step-by-step explanation:
To determine how fast a 40.0 kg wagon is moving after travelling up a hill without friction, we use the work-energy principle. The work done on the wagon by the tow rope is used to increase the wagon's kinetic energy. As there is no friction, the work done by the force of gravity as the wagon moves up the incline will be the only other work affecting the wagon's energy.
The work done by the tow rope (work = force x distance x cos(θ)) is 140 N multiplied by the distance traveled along the incline. We subtract the work done against gravity to find the net work, which equals the change in kinetic energy of the wagon (½mv²).
For (a), moving 30.0 m up the hill, the work done by the tow rope is:
Work = 140 N x 30.0 m x cos(0°) = 4200 J
For (b), moving 80.0 m up the hill:
Work = 140 N x 80.0 m x cos(0°) = 11200 J
The change in kinetic energy will be equal to the net work done. Therefore, by solving ½mv² = net work for 'v', we can find the speed of the wagon after each specified distance.