Connected particles via tow-bar on horizontal surface

A truck of mass 2400 kg is pulling a trailer of mass \(M\) kg along a straight horizontal road. The tow bar, connecting the truck to the trailer, is horizontal and parallel to the direction of motion. The tow bar is modelled as being light and inextensible. The resistance forces acting on the truck and the trailer are constant and of magnitude 400 N and 200 N respectively. The acceleration of the truck is 0.5 m s\(^{-2}\) and the tension in the tow bar is 600 N.

1. Find the magnitude of the driving force of the truck. [3]
2. Find the value of \(M\). [3]
3. Explain how you have used the fact that the tow bar is inextensible in your calculations. [1]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

1. the acceleration of the car and trailer, [3]
2. the magnitude of the tension in the towbar. [3]

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

1. find the value of \(F\). [7]

\includegraphics{figure_4} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at 20 m s\(^{-1}\) as they enter a zone where the speed limit is 14 m s\(^{-1}\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s\(^{-1}\) is 100 m.

1. Find the deceleration of the truck and the car. [3]

The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos \(\theta = 0.9\), find

1. the force in the towbar, [4]
2. the value of \(R\). [4]

A trailer of mass \(600\) kg is attached to a car of mass \(1100\) kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8\) m s\(^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is \(700\) N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is \(2100\) N, find the resistance to motion of the car. [3]

A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \text{ m s}^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is 2100 N, find the resistance to motion of the car. [3]

A car of mass 1.25 tonnes tows a caravan of mass 0.75 tonnes along a straight, level road. The total resistance to motion experienced by the car and the caravan is 1200 N. The car and caravan accelerate uniformly from rest to 25 m s\(^{-1}\) in 20 seconds.

1. Calculate the driving force produced by the car's engine. [4 marks]

Given that the resistance to motion experienced by the car and by the caravan are in the same ratio as their masses,

1. find these resistances and the tension in the towbar. [4 marks]

When the car and caravan are travelling at a steady speed of 25 m s\(^{-1}\), the towbar snaps. Assuming that the caravan experiences the same resistive force as before,

1. calculate the distance travelled by the caravan before it comes to rest, [5 marks]
2. give a reason why your answer to \((c)\) may be unrealistic. [2 marks]

A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.

1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]

When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,

1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
2. Show that tension in the tow-bar remains unchanged. [3 marks]

A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)

1. 1. Find R. [3 marks]
   2. Find the tension in the rope. [3 marks]
2. State a necessary assumption that you have made. [1 mark]
3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
   1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
   2. Explain the change in motion that the driver noticed. [2 marks]

Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics{figure_8} The acceleration of the system is 0.3 m s\(^{-2}\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). [2]

Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.

1. Calculate the new acceleration of the trucks. [2]
2. Calculate the force in the coupling between the trucks. [2]

Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics{figure_8} The acceleration of the system is 0.3 ms\(^{-2}\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). [2]

Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.

1. Calculate the new acceleration of the trucks. [2]
2. Calculate the force in the coupling between the trucks. [2]