Drawing forces for a car towing a caravan uphill

Expert intro: The inclined-road forces on the car and caravan must be separated carefully, because the coupling force, resistances, and weight components act on different bodies.

Detailed walkthrough

What each free-body diagram must show

For the car, the diagram should include the driving force of $9000\text{ N}$ acting up the slope, the resistance of $600\text{ N}$ acting down the slope, the tension or coupling force from the caravan acting down the slope on the car, the normal reaction perpendicular to the road, and the weight of the car acting vertically downward.

For the caravan, the diagram should include the coupling force from the car acting up the slope, the resistance of $1200\text{ N}$ acting down the slope, the normal reaction perpendicular to the road, and the weight of the caravan acting vertically downward.

How to label the forces correctly

The slope angle is $8^\circ$, so each weight vector may be resolved into components parallel and perpendicular to the road if the question or marking scheme expects that detail. The parallel component acts down the slope, and the perpendicular component presses into the road. The coupling between the car and caravan is light and parallel to the road, so the force in the coupling acts along the incline rather than at an angle.

The car and caravan are separate objects, so do not draw one combined force diagram unless the question specifically asks for the system as a whole. When drawing free-body diagrams, every force must act on exactly one body, and the arrow directions should show the direction of the force on that body, not the direction of motion.

Key modelling points for exam marks

A correct diagram for the car should show the driving force up the slope and the resistive force down the slope. The coupling force on the car points down the slope because the caravan pulls back on the car.

A correct diagram for the caravan should show the coupling force up the slope and the resistance down the slope. If the diagram includes weights resolved into components, the component parallel to the slope should be marked as $mg\sin 8^\circ$ and the perpendicular component as $mg\cos 8^\circ$ for each vehicle.

Common diagram mistakes

A frequent error is drawing the coupling force in the same direction on both vehicles. By Newton's third law, the force pair must be equal in size and opposite in direction: the car exerts a pull on the caravan up the slope, and the caravan exerts an equal pull back on the car down the slope.

Another common issue is mixing up resistance and driving force directions. Resistive forces always oppose the motion, so both resistance arrows should point down the slope here because the car and caravan are being pulled uphill. If the weights are left as arrows straight down, that is acceptable for a free-body diagram unless the examiner specifically requests resolved components.

A neat labelled sketch with all arrows clearly attached to the correct body is usually enough for full diagram marks, provided the directions are physically consistent.

💡 Pitfall guide

A frequent place this force-diagram question goes wrong is the coupling force between the car and caravan. Students often draw the same arrow direction on both vehicles, but Newton's third law requires opposite directions on the two bodies. The car feels a backward pull from the caravan, while the caravan feels a forward pull from the car. Another trap is putting the resistive force in the direction of travel because it is easy to think of resistance as something that helps the vehicle slow down. In a diagram, resistance must oppose the uphill motion, so it points down the slope for both objects. Do not combine the car and caravan into one body unless instructed; the exam wants separate free-body diagrams. Finally, if you decide to resolve weight into components, label the axes consistently with the slope or your marks can be lost even if the physics idea is right.

🔄 Real-world variant

If the road were level instead of being inclined at $8^\circ$, the free-body diagrams would change in a simple but important way. The weights would still act vertically downward, but there would be no need to resolve them into slope-parallel and slope-perpendicular components because there is no incline. The driving force on the car would still point forward, the resistance forces would still oppose the motion, and the coupling force would still act in opposite directions on the car and caravan. If the caravan were detached, the diagram for the car alone would no longer include a coupling force, and the car would have to balance its driving force against resistance and any acceleration effects. This variant shows that the force pattern depends on the geometry of the road and on whether the vehicles are connected.

🔍 Related terms

free-body diagram, Newton's third law, forces on incline