3.03c Newton's second law: F=ma one dimension

2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .

1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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 \(F\).

6 An elevator is pulled vertically upwards by a cable. The elevator accelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s , then travels at constant speed for 25 s . The elevator then decelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) until it comes to rest.

1. Find the greatest speed of the elevator and hence draw a velocity-time graph for the motion of the elevator.
2. Find the total distance travelled by the elevator.
   The mass of the elevator is 1200 kg and there is a crate of mass \(m \mathrm {~kg}\) resting on the floor of the elevator.
3. Given that the tension in the cable when the elevator is decelerating is 12250 N , find the value of \(m\).
4. Find the greatest magnitude of the force exerted on the crate by the floor of the elevator, and state its direction.

2 A particle \(P\) of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between \(P\) and the floor is \(\mu\). A force of magnitude 3 N is applied to \(P\) upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is initially at rest and accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time it takes for \(P\) to travel a distance of 1.44 m from its starting point.
2. Find \(\mu\).

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881} \(S _ { 1 }\) and \(S _ { 2 }\) are light inextensible strings, and \(A\) and \(B\) are particles each of mass 0.2 kg . Particle \(A\) is suspended from a fixed point \(O\) by the string \(S _ { 1 }\), and particle \(B\) is suspended from \(A\) by the string \(S _ { 2 }\). The particles hang in equilibrium as shown in the diagram.

1. Find the tensions in \(S _ { 1 }\) and \(S _ { 2 }\). The string \(S _ { 1 }\) is cut and the particles fall. The air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.2 N .
2. Find the acceleration of the particles and the tension in \(S _ { 2 }\).

6 A car of mass 1200 kg travels along a horizontal straight road. The power of the car's engine is 20 kW . The resistance to the car's motion is 400 N .

1. Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Show that the maximum possible speed of the car is \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done by the car's engine as the car travels from a point \(A\) to a point \(B\) is 1500 kJ .
3. Given that the car is travelling at its maximum possible speed between \(A\) and \(B\), find the time taken to travel from \(A\) to \(B\).

1 A car of mass 1200 kg travels on a horizontal straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that the car's speed increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 525 m , find the value of \(a\). The car's engine exerts a constant driving force of 900 N . The resistance to motion of the car is constant and equal to \(R \mathrm {~N}\).
2. Find \(R\).

4 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_568_1084_269_532} The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point \(A\) above the surface of liquid in a container. The downward velocity of the stone \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone hits the surface of the liquid with velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0.7\). It reaches the bottom of the container with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 1.2\).

1. Find
   1. the height of \(A\) above the surface of the liquid,
   2. the depth of liquid in the container.
   3. Find the deceleration of the stone while it is moving in the liquid.
   4. Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N , find the mass of the stone.

3 A car travels along a horizontal straight road with increasing speed until it reaches its maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and equal to \(R \mathrm {~N}\), and the power provided by the car's engine is 18 kW .

1. Find the value of \(R\).
2. Given that the car has mass 1200 kg , find its acceleration at the instant when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_702_709_269_719} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed peg. The particles are held at rest with the string taut. Both particles are at a height of 0.9 m above the ground (see diagram). The system is released and each of the particles moves vertically. Find

1. the acceleration of \(P\) and the tension in the string before \(P\) reaches the ground,
2. the time taken for \(P\) to reach the ground.

4 A car of mass 1230 kg increases its speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 24.5 s . The table below shows corresponding values of time \(t \mathrm {~s}\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Using the values in the table, find the average acceleration of the car for \(0 < t < 0.5\) and for \(16.3 < t < 24.5\). While the car is increasing its speed the power output of its engine is constant and equal to \(P \mathrm {~W}\), and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\).
2. Assuming that the values obtained in part (i) are approximately equal to the accelerations at \(v = 5\) and at \(v = 20\), find approximations for \(P\) and \(R\).

4 A train of mass 400000 kg is moving on a straight horizontal track. The power of the engine is constant and equal to 1500 kW and the resistance to the train's motion is 30000 N . Find

1. the acceleration of the train when its speed is \(37.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the steady speed at which the train can move.

3 A car has mass 800 kg . The engine of the car generates constant power \(P \mathrm {~kW}\) as the car moves along a straight horizontal road. The resistance to motion is constant and equal to \(R \mathrm {~N}\). When the car's speed is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the car's speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.33 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

6 A particle \(P\) of mass 0.2 kg is released from rest at a point 7.2 m above the surface of the liquid in a container. \(P\) falls through the air and into the liquid. There is no air resistance and there is no instantaneous change of speed as \(P\) enters the liquid. When \(P\) is at a distance of 0.8 m below the surface of the liquid, \(P\) 's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only force on \(P\) due to the liquid is a constant resistance to motion of magnitude \(R \mathrm {~N}\).

1. Find the deceleration of \(P\) while it is falling through the liquid, and hence find the value of \(R\). The depth of the liquid in the container is \(3.6 \mathrm {~m} . P\) is taken from the container and attached to one end of a light inextensible string. \(P\) is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of \(R \mathrm {~N}\) continues to act. The particle reaches the surface 4 s after leaving the bottom of the container.
2. Find the tension in the string.

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}

3 A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude 300 N .

1. Find, in kW , the rate at which the engine of the car is working when the car has a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at \(90 \%\) of the power found in part (i).

7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.

1. Given that the plane is smooth, find
   1. the acceleration of the particle,
   2. the change in kinetic energy after the particle has moved 12 m up the plane.
   3. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
      (a) Find the acceleration of the particle.
      (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.

2 A particle \(P\) of mass \(m \mathrm {~kg}\) moves on an arc of a circle with centre \(O\) and radius \(a\) metres. At time \(t = 0\) the particle is at the point \(A\). At time \(t\) seconds, angle \(P O A = \sin ^ { 2 } 2 t\). Show that the radial component of the acceleration of \(P\) at time \(t\) seconds has magnitude \(\left( 4 a \sin ^ { 2 } 4 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the value of \(t\) when the transverse component of the acceleration of \(P\) is first equal to zero,
2. the magnitude of the resultant force acting on \(P\) when \(t = \frac { 1 } { 12 } \pi\).

6 A particle of mass 3 kg falls from rest at a point 5 m above the surface of a liquid which is in a container. There is no instantaneous change in speed of the particle as it enters the liquid. The depth of the liquid in the container is 4 m . The downward acceleration of the particle while it is moving in the liquid is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the resistance to motion of the particle while it is moving in the liquid.
2. Sketch the velocity-time graph for the motion of the particle, from the time it starts to move until the time it reaches the bottom of the container. Show on your sketch the velocity and the time when the particle enters the liquid, and when the particle reaches the bottom of the container.

2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.

1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
2. Find the distance that the particle travels along the ground before it comes to rest.

3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .

1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .

5 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-3_355_1048_255_552} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\).
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.

6 A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant resistance to motion of 20 N . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the total mass of the cyclist and bicycle is 80 kg . The cyclist comes to a hill inclined at \(2 ^ { \circ }\) to the horizontal. When the cyclist starts climbing the hill, he increases his power to a constant 300 W . The resistance to motion remains 20 N .
2. Show that the steady speed up the hill which the cyclist can maintain when working at this power is \(6.26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the acceleration at an instant when the cyclist is travelling at \(90 \%\) of the speed in part (ii).

1 A block of mass 3 kg is initially at rest on a smooth horizontal floor. A force of 12 N , acting at an angle of \(25 ^ { \circ }\) above the horizontal, is applied to the block. Find the distance travelled by the block in the first 5 seconds of its motion.