3.03d Newton's second law: 2D vectors

5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at the bottom of the hill their speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. It is given that as the car and trailer descend the hill, the engine of the car does 150000 J of work, and there are no resistance forces. Find the length of the hill.
2. It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m , and the acceleration of the car and trailer is constant. Find the tension in the tow-bar between the car and trailer.

2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N . The power of the car's engine is 22500 W .

1. Show that the speed of the car is \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The car, moving with speed \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), comes to a section of the hill which is inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.

4 The total mass of a cyclist and her bicycle is 70 kg . The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). At an instant when the cyclist's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude \(F \mathrm {~N}\).

1. Find the value of \(F\).
2. Find the steady speed that the cyclist could maintain up the hill when working at this power. [2]

5 A car of mass 1600 kg travels at constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle of \(\sin ^ { - 1 } 0.12\) to the horizontal.

1. Find the change in potential energy of the car in 30 s .
2. Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.
3. Calculate, in kW , the power developed by the engine of the car.
4. Given that this power is suddenly decreased by \(15 \%\), find the instantaneous deceleration of the car.

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.

4 A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.

1. The car moves along a horizontal section of the road at a constant speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the work done against the resisting force during the first 8 seconds.
   2. Calculate, in kW , the power developed by the engine of the car.
   3. Given that this power is suddenly increased by 12 kW , find the instantaneous acceleration of the car.
2. The car now travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a section of the road inclined at \(\theta ^ { \circ }\) to the horizontal, with the engine working at 64 kW . Find the value of \(\theta\).

6 A car of mass 1750 kg is pulling a caravan of mass 500 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 650 N and 150 N respectively.

1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the power of the car's engine.
   2. The engine's power is now suddenly increased to 40 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-4_246_665_253_739} Two rectangular boxes \(A\) and \(B\) are of identical size. The boxes are at rest on a rough horizontal floor with \(A\) on top of \(B\). Box \(A\) has mass 200 kg and box \(B\) has mass 250 kg . A horizontal force of magnitude \(P\) N is applied to \(B\) (see diagram). The boxes remain at rest if \(P \leqslant 3150\) and start to move if \(P > 3150\).

1. Find the coefficient of friction between \(B\) and the floor. The coefficient of friction between the two boxes is 0.2 . Given that \(P > 3150\) and that no sliding takes place between the boxes,
2. show that the acceleration of the boxes is not greater than \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. find the maximum possible value of \(P\).

1 A car of mass 700 kg is travelling along a straight horizontal road. The resistance to motion is constant and equal to 600 N .

1. Find the driving force of the car's engine at an instant when the acceleration is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the car's speed at this instant is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the rate at which the car's engine is working.

3 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-2_748_1410_979_370} The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:

• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.
  1. Show that the total distance fallen is 1048 m .

The weight of the parachutist is 850 N .

Find the upward force on the parachutist due to the parachute, during the second stage.

5 \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-4_620_623_255_760} A small block of mass 1.25 kg is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle \(\theta\) is such that \(\cos \theta = 0.28\) and \(\sin \theta = 0.96\). A horizontal frictional force also acts on the block, and the block is in equilibrium.

1. Show that the magnitude of the frictional force is 7.5 N and state the direction of this force.
2. Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. The force of magnitude 6.1 N is now replaced by a force of magnitude 8.6 N acting in the same direction, and the block begins to move.
3. Find the magnitude and direction of the acceleration of the block.

5 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_529_195_255_977} A block \(A\) of mass 3 kg is attached to one end of a light inextensible string \(S _ { 1 }\). Another block \(B\) of mass 2 kg is attached to the other end of \(S _ { 1 }\), and is also attached to one end of another light inextensible string \(S _ { 2 }\). The other end of \(S _ { 2 }\) is attached to a fixed point \(O\) and the blocks hang in equilibrium below \(O\) (see diagram).

1. Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\). The string \(S _ { 2 }\) breaks and the particles fall. The air resistance on \(A\) is 1.6 N and the air resistance on \(B\) is 4 N .
2. Find the acceleration of the particles and the tension in \(S _ { 1 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-4_333_1001_264_573} A light inextensible string of length 5.28 m has particles \(A\) and \(B\), of masses 0.25 kg and 0.75 kg respectively, attached to its ends. Another particle \(P\), of mass 0.5 kg , is attached to the mid-point of the string. Two small smooth pulleys \(P _ { 1 }\) and \(P _ { 2 }\) are fixed at opposite ends of a rough horizontal table of length 4 m and height 1 m . The string passes over \(P _ { 1 }\) and \(P _ { 2 }\) with particle \(A\) held at rest vertically below \(P _ { 1 }\), the string taut and \(B\) hanging freely below \(P _ { 2 }\). Particle \(P\) is in contact with the table halfway between \(P _ { 1 }\) and \(P _ { 2 }\) (see diagram). The coefficient of friction between \(P\) and the table is 0.4 . Particle \(A\) is released and the system starts to move with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the part \(A P\) of the string is \(T _ { A } \mathrm {~N}\) and the tension in the part \(P B\) of the string is \(T _ { B } \mathrm {~N}\).

1. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(a\).
2. Show by considering the motion of \(P\) that \(a = 2\).
3. Find the speed of the particles immediately before \(B\) reaches the floor.
4. Find the deceleration of \(P\) immediately after \(B\) reaches the floor. \end{document}

2 A car of mass 1250 kg travels up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.02\). The power provided by the car's engine is 23 kW . The resistance to motion is constant and equal to 600 N . Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of \(P \mathrm {~W}\) while moving on a straight road which is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = 0.1\). When moving uphill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When moving downhill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is \(R \mathrm {~N}\). Find the values of \(P\) and \(R\).

5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .

1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A cyclist, working at a constant rate of 400 W , travels along a straight road which is inclined at \(2 ^ { \circ }\) to the horizontal. The total mass of the cyclist and his cycle is 80 kg . Ignoring any resistance to motion, find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling

1. uphill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. downhill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 Particles \(A\) of mass 0.65 kg and \(B\) of mass 0.35 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(B\) is held at rest with the string taut and both of its straight parts vertical. The system is released from rest and the particles move vertically. Find the tension in the string and the magnitude of the resultant force exerted on the pulley by the string.

1 A racing cyclist, whose mass with his cycle is 75 kg , works at a rate of 720 W while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to \(R \mathrm {~N}\).

1. Given that the cyclist is accelerating at \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when his speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(R\).
2. Given that the cyclist's acceleration is positive, show that his speed is less than \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.

1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
2. Find the height of the particles above the floor at the instant that they started to move.

3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
2. Find how far up the hill the cyclist travels before coming to rest.

2 A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a point \(A\) and has a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches a point \(B\). The distance \(A B\) is 15 m . The pulling force has magnitude 220 N and acts at an angle of \(\alpha ^ { \circ }\) above the horizontal. The work done against the resistance to motion acting on the box, as the box moves from \(A\) to \(B\), is 3000 J . Find the value of \(\alpha\).

3 The resistance to motion acting on a runner of mass 70 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the runner's speed and \(k\) is a constant. The greatest power the runner can exert is 100 W . The runner's greatest steady speed on horizontal ground is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 6.25\).
2. Find the greatest steady speed of the runner while running uphill on a straight path inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).

7 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-4_492_1365_255_392} An elevator is pulled vertically upwards by a cable. The velocity-time graph for the motion is shown above. Find

1. the distance travelled by the elevator,
2. the acceleration during the first stage and the deceleration during the third stage. The mass of the elevator is 800 kg and there is a box of mass 100 kg on the floor of the elevator.
3. Find the tension in the cable in each of the three stages of the motion.
4. Find the greatest and least values of the magnitude of the force exerted on the box by the floor of the elevator.

6 A lorry of mass 12500 kg travels along a road from \(A\) to \(C\) passing through a point \(B\). The resistance to motion of the lorry is 4800 N for the whole journey from \(A\) to \(C\).

1. The section \(A B\) of the road is straight and horizontal. On this section of the road the power of the lorry's engine is constant and equal to 144 kW . The speed of the lorry at \(A\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration at \(B\) is \(0.096 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the acceleration of the lorry at \(A\) and show that its speed at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. The section \(B C\) of the road has length 500 m , is straight and inclined upwards towards \(C\). On this section of the road the lorry's driving force is constant and equal to 5800 N . The speed of the lorry at \(C\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the height of \(C\) above the level of \(A B\).