Newton's laws of motion

A force of magnitude F acts on a box of mass m that moves along a frictionless slope. The slope is at an angle α to the horizontal and the force acts at an angle β to the slope.

What is the magnitude of the acceleration of the block along the slope?

A man has a mass of 75.0 kg. He stands on weighing scales in a lift that accelerates upwards at 2.60 m s⁻².

What is the reading on the scales during the acceleration?

In 2021 the world land speed record was 1230 km h⁻¹. This was the average speed achieved by a jet-powered car in two runs. Each run was measured over a distance of 1.61 km.

The average speed for one of these runs was 343 m s⁻¹.

Calculate, in s, the time taken for the car to complete the other run.

Engineers are designing a new jet-powered car to break this record.

Figure 1 shows the variation of speed with distance for the car, as predicted by the engineers.

The car reaches its maximum acceleration when it is 5600 m from the start. At this point the mass of the car is 6.50 × 10³ kg.

Determine the kinetic energy of the car at its maximum acceleration.

At any point on the graph in Figure 1, the acceleration is given by:

acceleration = speed × gradient of line

When the car is at its maximum acceleration, the power input to the jet engines is 640 MW.

Calculate the percentage of the input power used to accelerate the car at its maximum acceleration.

Scientists recommend that the average deceleration of the driver of the car should be less than 3 g.

Deduce whether the average deceleration is less than 3 g.

An object is moving in a straight line. A graph is plotted to show the variation of the momentum of the object with time.

Which quantities can be calculated from the gradient of the graph and the area under

A trolley moves down a slope with constant acceleration. The mass of the trolley is doubled and the trolley moves down the same slope again. Air resistance and friction are negligible.

Which is correct?

A teacher sets up a demonstration to show the relationship between circular motion and simple harmonic motion (SHM). She places a block on a turntable at a point 0.25 m from its centre, as shown in Figure 9.

The turntable rotates with an angular speed of 1.8 rad s^–1 and the block does not slip.

Calculate the time taken for the turntable to complete one revolution.

Figure 10 shows a plan view of the turntable and block. The turntable rotates in a clockwise direction.

Draw an arrow on Figure 10 to show the direction of the resultant force on the block.

The mass of the block is 0.12 kg.

Calculate the magnitude of the resultant force on the block.

Describe, with reference to one of Newton’s laws of motion, the evidence that a resultant force is acting on the block.

The teacher adjusts the angular speed of the turntable so that the block completes one rotation every 2.50 s. She sets up a simple pendulum above the centre of the turntable so that it swings in phase with the movement of the block.

Calculate the length of the simple pendulum.

A lamp is used to project shadow images of the block and pendulum bob on a screen. Both shadows appear to move with SHM across the screen. Figure 11 shows the images on the screen at one instant.

Initially the shadows move in phase with the same amplitude.

Air resistance affects the motion of the pendulum.

Suggest the effect this has on the amplitude relationship and the phase relationship between the moving shadows.

A truck of mass 2.1 × 10³ kg tows a car of mass 1.3 × 10³ kg along a horizontal road. The total resistive force on the car is 1100 N. The acceleration of the car and truck is 2.3 m s⁻².

What is the tension in the tow rope?

A parachutist descends to the ground at a constant speed with the parachute open.

Which force, together with the parachutist’s weight, makes a pair according to Newton’s third law of motion?

A Formula 1 racing car uses up its fuel during the race, causing its lap times to decrease.

The lap times decrease because

A pair of cameras is used on a motorway to help determine the average speed of vehicles travelling between the two cameras.

Figure 5 shows the speed–time graph for a car moving between the two cameras.

The speed limit for the motorway between the two cameras is 22 m s⁻¹.

Determine whether the average speed of the car exceeded this speed limit.

Markings called chevrons are used on motorways.

The chevron separation is designed to give a driver time to respond to any change in speed of the car in front. The driver is advised to keep a minimum distance d behind the car in front, as shown in Figure 6.

Government research suggests that the typical time for a driver to respond is between 1.6 s and 2.0 s.

Suggest a value for d where the speed limit is 31 m s⁻¹.

The chevron separation is based on the response time, not on the time taken for a car to stop.

The brakes of a car are applied when its speed is 31 m s⁻¹ and the car comes to rest. The total mass of the car is 1200 kg.

The average braking force acting on the car is 6.8 kN.

Calculate the time taken for the braking force to stop the car and the distance travelled by the car in this time.

Suggest why the chevron separation on motorways does not take into account the distance travelled as a car comes to rest after the brakes are applied.

At bends on motorways the road is sloped so that a car is less likely to slide out of its lane when travelling at a high speed.

Figure 7 shows a car of mass 1200 kg travelling around a curve of radius 200 m. The motorway is sloped at an angle of 5.0°.

Figure 8 shows the weight W and reaction force N acting on the car. The advisory speed for the bend is chosen so that the friction force down the slope is zero.

Suggest an appropriate advisory speed for this section of the motorway.

Safety barriers are used on UK motorways to prevent vehicles crossing from one carriageway to the other carriageway. The barriers also absorb some of the kinetic energy of a vehicle and deflect vehicles along the barrier.

The standard test of a safety barrier uses a vehicle that contains dummies. The total mass of the vehicle and its contents is 1.5 × 10³ kg and its initial speed is 110 km h^–1

Show that the initial kinetic energy of the test vehicle is 700 kJ.

The test vehicle hits a steel safety barrier at an angle of 20°, as shown in Figure 7.

Calculate the component of the momentum of the test vehicle in a direction along the line of the safety barrier. Give an appropriate unit for your answer.

Immediately after the collision, the test vehicle moves along the safety barrier with no change in its momentum in this direction.

Show that the kinetic energy lost in the collision is about 80 kJ.

The steel safety barrier deforms during the collision. For the barrier to pass the test, the test vehicle should not move more than 1.5 m towards the other carriageway.

The barrier can apply an average force of 60 kN at right angles to the carriageway.

Deduce whether the safety barrier will pass the test.

A different safety barrier uses a solid concrete wall which does not deform. The same standard test is carried out on a concrete wall.

Discuss which type of barrier would cause less damage to the dummies in the test.

A 0.20 kg mass is suspended from a spring. A 0.10 kg mass is suspended from the 0.20 kg mass using a thread of negligible mass. The system is in equilibrium and the thread is then cut.

What is the upward acceleration of the 0.20 kg mass at the instant that the thread is cut?

A lift of mass M is suspended from a cable. The lift descends with a downward acceleration, a. A frictional force F acts on the lift.

What is the tension T in the cable?

Horizontal escape lanes made of loose gravel have been constructed at the side of some roads on steep hills so that vehicles can stop safely when their brakes fail.

Figure 1 shows an engineer’s prediction of how the speed of an unpowered vehicle of mass 1.8 × 10⁴ kg will vary with time as the vehicle comes to rest in an escape lane.

Determine the force decelerating the vehicle 2.0 s after entering the escape lane.

Deduce whether a lane of length 85 m is long enough to stop the vehicle, assuming that the engineer’s graph is correct.

Discuss the energy transfers that take place when a vehicle is decelerated in an escape lane.

An alternative to an escape lane containing gravel is an escape lane that consists of a ramp. An escape ramp is a straight road with a concrete surface that has a constant upward gradient.

One escape ramp makes an angle of 25° to the horizontal and is 85 m long.

Deduce whether this escape ramp is sufficient to stop the vehicle.

Assume that any frictional forces and air resistance that decelerate the vehicle are negligible.

Discuss whether an escape lane containing gravel or an escape ramp would provide the safer experience for the driver of the vehicle as it comes to rest.

Immediately after take-off from the surface of the Earth, a rocket of mass 12 000 kg accelerates vertically upwards at 1.4 m s⁻²

What is the thrust produced by the rocket motor?

Figure 8

The model consists of two identical trolleys of mass M on a ramp which is at 35° to the horizontal. The trolleys are connected by a wire that passes around a pulley of negligible mass at the top of the ramp.

Two concrete blocks each of mass m are loaded onto trolley A at the top of the ramp. The trolley is released and accelerates to the bottom of the ramp where it is stopped by a flexible buffer. The blocks are unloaded from trolley A and two blocks are loaded onto trolley B that is now at the top of the ramp. The trolleys are released and the process is repeated.

Figure 9 shows the side view of trolley A when it is moving down the ramp.

The tension in the wire when the trolleys are moving is T.

Draw and label arrows on Figure 9 to represent the magnitudes and directions of any forces and components of forces that act on trolley A parallel to the ramp as it travels down the ramp.

Assume that no friction acts at the axle of the pulley or at the axles of the trolleys and that air resistance is negligible.

Show that the acceleration a of trolley B along the ramp is given by

Compare the momentum of loaded trolley A as it moves downwards with the momentum of loaded trolley B.

In practice, for safety reasons there is a friction brake in the pulley that provides a resistive force to reduce the acceleration to 25% of the maximum possible acceleration.

The distance travelled for each journey down the ramp is 9.0 m.

The following data apply to the arrangement.

Mass of a trolley M = 95 kg Mass of a concrete block m = 30 kg

Calculate the time taken for a loaded trolley to travel down the ramp.

It takes 12 s to remove the blocks from the lower trolley and reload the upper trolley.

Calculate the number of blocks that can be transferred to the lower level in 30 minutes.