Inclined road towing

5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
   On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
2. 1. Find the acceleration of the van and the tension in the tow-bar.
   2. Find the speed of the van when it has travelled a distance of 375 m up the hill.

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 14 }\). The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

1. show that the acceleration of the system is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
2. find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
3. Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
4. Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.