Car towing trailer, horizontal

2 A car of mass 1500 kg is towing a trailer of mass \(m \mathrm {~kg}\) along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of \(F \mathrm {~N}\) on the car and a resistance force of 200 N on the trailer. The driving force of the car's engine is 3200 N , the acceleration of the car is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the tow-bar is 300 N . Find the value of \(m\) and the value of \(F\).

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.

1. Find the magnitude of the force in the tow-bar.
2. Find the braking force.
3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.

2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.

1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
   The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
2. Find the least possible value of the braking force which will cause the rope to become slack.

2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),

1. show that \(R = 860\),
2. find the tension in the tow-bar.

6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find

1. the magnitude of the acceleration of the van and the car,
2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.

6. A car pulls a trailer along a straight horizontal road using a light inextensible towbar. The mass of the car is \(M \mathrm {~kg}\), the mass of the trailer is 600 kg and the towbar is horizontal and parallel to the direction of motion. There is a resistance to motion of magnitude 200 N acting on the car and a resistance to motion of magnitude 100 N acting on the trailer. The driver of the car spots a hazard ahead. Instantly he reduces the force produced by the engine of the car to zero and applies the brakes of the car. The brakes produce a braking force on the car of magnitude 6500 N and the car and the trailer have a constant deceleration of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Given that the resistances to motion on the car and trailer are unchanged and that the car comes to rest after travelling 40.5 m from the point where the brakes were applied, find

1. the thrust in the towbar while the car is braking,
2. the value of \(M\),
3. the time it takes for the car to stop after the brakes are applied.

7. A truck of mass 1600 kg is towing a car of mass 960 kg along a straight horizontal road. The truck and the car are joined by a light rigid tow bar. The tow bar is horizontal and is parallel to the direction of motion. The truck and the car experience constant resistances to motion of magnitude 640 N and \(R\) newtons respectively. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitude of the acceleration of the truck and the car is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 436\)
2. Find the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 15 }\). The truck and the car move down a line of greatest slope of the hill with the tow bar parallel to the direction of motion. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitudes of the resistances to motion on the truck and the car are 640 N and 436 N respectively.
3. Find the magnitude of the acceleration of the truck and the car as they move down the hill.
   
   \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-27_67_59_2654_1886}

5 A car is towing a trailer along a straight road using a light tow-bar which is parallel to the road. The masses of the car and the trailer are 900 kg and 250 kg respectively. The resistance to motion of the car is 600 N and the resistance to motion of the trailer is 150 N .

1. At one stage of the motion, the road is horizontal and the pulling force exerted on the trailer is zero.
   1. Show that the acceleration of the trailer is \(- 0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the driving force exerted by the car.
   3. Calculate the distance required to reduce the speed of the car and trailer from \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   4. At another stage of the motion, the car and trailer are moving down a slope inclined at \(3 ^ { \circ }\) to the horizontal. The resistances to motion of the car and trailer are unchanged. The driving force exerted by the car is 980 N . Find
      (a) the acceleration of the car and trailer,
      (b) the pulling force exerted on the trailer.

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

1. The car and trailer are travelling at constant speed.
2. The car and trailer have acceleration \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A car and its trailer travel along a straight, horizontal road. The coupling between them is light and horizontal. The car has mass 900 kg and resistance to motion 100 N , the trailer has mass 700 kg and resistance to motion 300 N , as shown in Fig. 4. The car and trailer have an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Calculate the driving force of the car.
2. Calculate the force in the coupling.

4. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.
The rope is horizontal and parallel to the direction of motion of the car.

• The resistance to motion of the car is modelled as a constant force of magnitude \(2 R\) newtons
• The resistance to motion of the trailer is modelled as a constant force of magnitude \(R\) newtons
• The rope is modelled as being light and inextensible
• The acceleration of the car is modelled as \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N . Using the model,

1. find the value of \(a\) In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be \(a _ { 1 } \mathrm {~ms} ^ { - 2 }\)
2. State how the value of \(a _ { 1 }\) compares with the value of \(a\)
3. State one limitation of the model used for the resistance to motion of the car.

4. \begin{figure}[h] \end{figure} Figure 2 shows a car towing a trailer along a straight horizontal road.
The mass of the car is 800 kg and the mass of the trailer is 600 kg .
The trailer is attached to the car by a towbar which is parallel to the road and parallel to the direction of motion of the car and the trailer. The towbar is modelled as a light rod.
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer is modelled as a constant force of magnitude R newtons. The engine of the car is producing a constant driving force that is horizontal and of magnitude 1740 N. The acceleration of the car is \(0.6 \mathrm {~ms} ^ { - 2 }\) and the tension in the towbar is T newtons.
Using the model,

1. show that \(\mathrm { R } = 500\)
2. find the value of T . At the instant when the speed of the car and the trailer is \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks.
   The trailer moves a further distance d metres before coming to rest.
   The resistance to the motion of the trailer is modelled as a constant force of magnitude 500 N. Using the model,
3. show that, after the towbar breaks, the deceleration of the trailer is \(\frac { 5 } { 6 } \mathrm {~ms} ^ { - 2 }\)
4. find the value of d. In reality, the distance d metres is likely to be different from the answer found in part (d).
5. Give two different reasons why this is the case.

2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine of the car produces a driving force of 6000 N . A resistance of 800 N acts on the car. A resistance of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.

1. Draw a force diagram showing all the horizontal forces on the car and the trailer.
2. Calculate the acceleration of the car and trailer.

5 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal towbar. A resistance force of magnitude \(R\) newtons acts on the car and a resistance force of magnitude \(2 R\) newtons acts on the caravan. The car and caravan accelerate at a constant \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when a driving force of magnitude 4720 newtons acts on the car.

1. Find \(R\).
2. Find the tension in the towbar.

1 A car of mass 1000 kg is travelling along a straight, level road. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the car when a resultant force of 2000 N acts on it in the direction of its motion. How long does it take the car to increase its speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(12.5 \mathrm {~ms} ^ { - 1 }\) ? The car has an acceleration of \(1.4 \mathrm {~ms} ^ { - 2 }\) when there is a driving force of 2000 N .
2. Show that the resistance to motion of the car is 600 N . A trailer is now atached to the car, as shown in Fig. 6.2. The car still has a driving force of 2000 N and resistance to motion of 600 N . The trailer has a mass of 800 kg . The tow-bar connecting the car and the trailer is light and horizontal. The car and trailer are accelerating at \(0.7 \mathrm {~ms} ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-1_165_883_1279_554} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Show that the resistance to the motion of the trailer is 140 N .
4. Calculate the force in the tow bar. The driving force is now removed and a braking force of 610 N is applied to the car. All the resistances to motion remain as before. The trailer has no brakes.
5. Calculate the new acceleration. Calculate also the force in the tow-bar, stating whether it is a tension or a thrust (compression).

4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}

1. Find the tension in the tow rope.
2. Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
3. In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.

15 A cyclist is towing a trailer behind her bicycle. She is riding along a straight, horizontal path at a constant speed. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-22_371_723_447_657} A tension of \(T\) newtons acts on the connecting rod between the bicycle and the trailer.
The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle. The constant resistance force acting on the trailer is 12 N
15

1. State the value of \(T\) giving a clear reason for your answer.
   15
2. State one assumption you have made in reaching your answer to part (a).

A car, of mass 1800 kg, pulls a trailer of mass 350 kg along a straight horizontal road. When the car is accelerating at \(0.2\) ms\(^{-2}\), the resistances to the motion of the car and trailer have magnitudes 300 N and 100 N respectively. Find, at this time,

1. the driving force produced by the engine of the car, [3 marks]
2. the tension in the tow-bar between the car and the trailer. [4 marks]

A tractor and its driver have a combined mass of \(m\) kilograms. The tractor is towing a trailer of mass \(4m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of \(11080 \text{N}\) and a total resistance force of \(160 \text{N}\) act on the tractor. A total resistance force of \(600 \text{N}\) acts on the trailer. The tractor and the trailer have an acceleration of \(0.8 \text{m s}^{-2}\)

1. Find \(m\). [3 marks]
2. Find the tension in the tow bar. [2 marks]
3. At the instant the speed of the tractor reaches \(18 \text{km h}^{-1}\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \text{km h}^{-1}\) [4 marks]

\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$

1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
   1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
   2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
3. State one assumption you have made about the tow bar. [1 mark]

A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \text{ m s}^{-2}\) along a straight horizontal road as shown in the diagram below. \includegraphics{figure_18} The van has a total mass of 2780 kg The car has a total mass of 1620 kg The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons. The car experiences a total resistance force of \(0.6R\) newtons.

1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = kD - 18$$ where \(k\) is a constant. Find \(k\) [5 marks]
2. State one assumption that must be made in answering part (a). [1 mark]

A car of mass \(1200 \text{ kg}\) pulls a trailer of mass \(400 \text{ kg}\) along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of \(3200 \text{ N}\). The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \text{ m s}^{-2}\),

1. find the resistance to motion on the trailer, [4]
2. find the tension in the tow-rope. [3]

When the car and trailer are travelling at \(25 \text{ m s}^{-1}\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,

1. find the distance the trailer travels before coming to a stop, [4]
2. state how you have used the modelling assumption that the tow-rope is inextensible. [1]